OSX saves INVISIBLE FILES by default..??

Could they be saved in your user account's Document folder?
Some items by default, will go to an assigned or assumed location.

There also is a way to make 'really invisible files' appear in the Finder.
But what you are saying does not jibe with the idea of doing this task.

There is a tool in OnyX that allows one to make all invisible files visible;
this would be used to see unix or coded content not normally seen; and
not for the purposes of recovering items not seen in Finder.

The files you say are Invisible... Are these items supposed to be in Finder
as a result of using a Download Manager in a browser? You can assign
the location where items would be saved to, so perhaps they are going to
somewhere other than the Finder.

A second-view of the Finder (each user account has a Desktop folder) may
allow you to see any content there in a form other than by-icon. If it is there
and not visible, you still should be able to see it. Or force it to be seen. But
the real question would be, why would anything saved in Finder be invisible?

By default, it wouldn't be invisible. The odds are, the items aren't in the Finder
or the user's Desktop Folder. (OS9 has a different creature named Finder.)

If I had invisible items in the Finder or Desktop folder, I'd use my OnyX utility
and make everything there visible. For a short while, then revert it to normal.
OnyX is a free download the from Titanium Software's web site.

There must be a reason why the issue you are experiencing, is occurring in
your system; either settings are messed up, permissions or privileges, or a
problem in the OS X in your configuration. I've never had that happen in any
of the OS X versions I've owned. (From Beta, to 10.0/10.1/10.2/10.3/10.4/10.5)
So, that this is happening in your situation does bring up other questions.

If you perform basic preventative maintenance, to include booting in SafeBoot
(shift key held down through startup, login, then desktop appears) you could
run Disk Utility there, to 'repair disk permissions' on Macintosh HD, then restart
the computer normally. This may help; but perhaps not if there is a source of
unusual corruption in the way the computer's data files are handled.

Does the system's Console utility show anything unusual in the Log files there?
Most system activity and resulting errors or logs should be available to you.

Hopefully someone else will have the smarts to ask the correct ones.
Good luck & happy computing! 🙂

Did the files disappear recently or has this problem been going on since you began using os x?
What type of account do you have? A managed account may cause this situation. You want a standard account or an administrator account. Look in your account preferences. Look under parental controls.

Try locating the file with one of these programs.
I have found that spotlight is inconsistent when searching by filename. To search by filename, try:
FindAnyFile
"This is a free program for Mac OS 10.4 and later that lets you search for files on your disks, primarily on HFS formatted ones."
http://apps.tempel.org/FindAnyFile/

EasyFind
An alternative to or supplement of Spotlight and finds files, folders or contents in any file without the need for indexing.
http://www.apple.com/downloads/macosx/systemdiskutilities/easyfind.html

See this thread, maybe you turned on desktop hiding by accident:
http://discussions.apple.com/thread.jspa?threadID=2343275&tstart=0

OK, I think I see your problem, but first of all let me mention one thing, OSX isn't like OS9 you really shouldn't save files to the Desktop because OSX treats each file on the Destop as a special Quartz window, hence using up more Resources like CPU, GPU, RAM, etc., much better to save them to another folder.

Anyway, the problem is that Finder doesn't update itself regularly, but this'll help with that, Refresh Finder...

http://www.versiontracker.com/dyn/moreinfo/macosx/33066

If all of this issue is based on over-loading the Finder (in OS X this is not
the versatile super desktop it was under OS 9.2.2) then you should think
about doing something along the lines of what I do with anything that may
be of interest to me, while in my main daily user account...

On the root level of the hard disk drive, I have a folder; in this, are other
nested folders where items I have saved: including web page URLs not
bookmarked in a browser (saved to desktop by dragging out of browser),
miscellaneous images, text, documents - to include .PDFs of web pages,
and other items, by kind. These are all in one folder, I put an alias on
the desktop in my user account; then to see all of this junk, I double-click
on the alias and presto: the junk bin of my organized saved stuff, opens.

I've randomly named this, Misc Extras folder; where other folders reside.
If I want to access some single saved item for awhile, temporarily, from
it, I may make an alias to that item, and put that on the desktop so I can
open it. I have an Apple Support tech .PDF file's alias out where I can
read through the dozens of pages; when done, I can trash the alias from
the desktop. The real file is hidden in the main folder. I may have up to 5
icons on the desktop at one time; one is the Mac HDD. Others are items
I am using this week.

{Since I edit digital images somewhat manually on the desktop, I keep it
mostly empty of extra debris; and do not use iPhoto to handle images.
So, every item is manually dated and rotated, saved, resized, in folders.
And they do not stay on the desktop; they are organized in dated folders.
Later on, these image folders are organized in a final fashion and burned
to a few DVDs, then get them all off the computer's hard disk drive.}

And on first blush, one folder opens to show folders. I can organize each of
them according to the View, icon size, Kind, Date, etc. This helps keep track.

Also, for extra applications and utilities that I may not want to keep an
icon link to in the Dock, I have an Saved Alias folder of aliases to those
apps or utilities I may want faster access to; instead of opening Apps
Folder from the hard disk drive. This folder also hides in the hard disk
drive, and perhaps even inside the same folder where all that desktop
junk bin stuff also hides. By dragging this Saved Alias folder into the
Dock, near the Trash can, a link to that folder stays in there. So, I do
not have Aliases to Apps on the Desktop, they are tidy in a folder, too.

The Dock is already full enough; with some extra stuff stuck in there.

Since I started organizing junk on the desktop back in OS6.0, this was
not a hard leap for me to keep doing what worked. What made me re-
think about keeping up on this idea was having seen other OS X users
whose desk space was cluttered and not even organized by alphabet,
size, kind or anything. Just over-lapping icons with no organization.
And these persons complained about the slow acting Finder desktop
window in OS X. Yet failed to act on advice to clean up the cause of it.

So, if organizing the desktop in OS X helps, and you keep track of
where the stuff ends up, that would be good. Some kinds of ideas
will only work well in the one user account; other ideas should work
in other user accounts too; but there may be ownership issues, then.

This is a little different than files just disappearing; or being saved as
invisible, by default. These items are categories apart, actually.

You can Force-Quit the Finder and relaunch it, that will refresh it;
but if there are dozens or hundreds of items stored there, that is
only a temporary shuffle and the answer is to organize it elsewhere.

And reasonable periodic and general maintenance, with the idea of
doing so as prevention, could help the issue you face with Finder
and files going missing. Maybe a bit more than 'repair disk' and a
'repair disk permissions' or SafeBoot, run Disk Utility's permission
repair and restart. Things such as AppleJack &/or OnyX can help.

Plus keeping track of remaining drive free-space. To archive stuff
off the computer's internal hard drive, to an external; to backup.

Good luck & happy computing! 🙂

{ edited }

This sounds like what happens if you have a server set up and your account is set up as a Mobile Account. Essentially, when you're saving your files, you're saving them to the copy of your home folder on the server, not on the local Mac. When you look at your desktop on the Mac, they don't show up because they weren't actually saved there, but rather on the home folder on the server.

The reason they show up the following day is that your Mac has had a chance to sync with the server and so those missing files are copied to the desktop on the local computer. Make sense?

You can verify this pretty easily. If you open an app and choose Save as..., and you click on the menu to select the folder you're saving in, it should indicate the path to that home folder you're in. If it's the right home folder, then it should be [Computer name]>[Startup drive name]>Users>[home folder name]. If it's saving to the server, then it'll be something like [Computer name]>Network>Servers>[server name]>etc.... I hope I didn't make that more confusing than it needs to be.

The default save location generally gets messed up like this when you have a Mobile Account and you switch computers. Without know more about this particular case, that would be my guess as to what happened recently. I can't think of how to correct that off the top of my head...I suspect it involves trashing a preferences file somewhere, but I'm sure it'll be mentioned here in the forums somewhere, unless someone else can chime in with that fix?