First, connecting to the network, any network, does allow other users on that same network to communicate with your computer. But, by default they will not be able to get into your computer unless you have allowed them to, by sharing files or folders for example. You can go to System Preferences -> Sharing to see what you have shared on your computer.
What is the remote control application that your tech installed? Most of the remote management tools out there require an account of some kind to either access the remote management console, and/or will require a username and password to access the computer itself. This might be something that you would ask your tech guy about.
To be absolutely sure there has not been some remote control software installed, you would have to scour the system for any software you do not recognize. You can start by looking in the Applications folder. Also run Activity Monitor and look for any processes that seem suspicious. Unless you are quite familiar with OS X, you will see a lot of processes running that you do not recognize, so this might not be a rewarding activity for you. If you trust your tech guy, I would suggest asking him to review your machine and making sure it is secure.
Lastly, to check access to your files, open Finder and go to the folder which contains the files you wish to check. Change to the list view (the icon with the horizontal lines), and look at the "Last Opened" column. This will tell you the last time the file was opened. Aside from that, there's not much you can do to see if they have ever been copied or opened. If all the previous steps have been resolved to your satisfaction, then this last step is probably not something you need to be worried about anyway.
Your best protections going forward are:
- Choose strong, complex passwords for ALL of your accounts (your computer, email, web accounts, this forum), and change them regularly. Unless you have a remarkable memory or very few accounts, you will need some piece of software or a notepad to keep track of them all.
- Don't use the same password for all of your accounts.
- Don't download or install software from anyone you don't know. If you're not sure, do a search on the software fist to see if anyone has had problems with it.
- Don't follow links in emails unless they are from a source you know. Even if it seems to be from some company you are familiar with, if you see strange grammar or lots of mispelled words, it's probably a fake.
- Don't give access to your computer to anyone you don't trust.
Hope this helps.