Hi,
For your "saved" pictures that are sent look in ~/Library/Messages/Attachments
This folder is where incoming File Transfers including pics are kept.
Open a Finder Window
Use the Go Menu > Go to Folder
Type (Copy and Paste) the above location into the Dialogue box
If recent pictures are not there it sounds like your computer is not writing them to that location.
Go to the View Menu and Show Path Bar.
Now click on the Messages icon in the Path Bar (Bottom of window)
Right Click the Attachments Folder and choose Get Info.
In the info Panel scroll to the bottom.
It should list your account Name and it should say Read and Write.
Highlight this Line.
Unlock the panel.
Use the Cogwheel like icon at the bottom edge.
This should say "Make yourname (me) the Owner" but it should be greyed out if you already own the Folder.
If it does not make yourself the Owner.
How things get written here is dependent on another route.
In View Menu > Go to Folder enter in ~/Library/Containers/com.apple.iChat/Data/Library/Messages
In here should be an Alias that open the ~/Library/Messages/Attachments folder (check the Path Bar to confirm)
Use the Back arrow to return to the ... Data/Library/Messages folder.
Click on the Data icon and the check you Own the Library folder and have Read and Write permissions.
If you need to change things within the Containers Folder then use this Resolve issues caused by changing the permissions of items in your home folder - Apple Support as it is likely you have lost Ownership or Permissions on more than one folder and the files they contain.

7:43 pm Thursday; August 9, 2018
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
MacBookPro 15" 2016 (High Sierra 10.13.x)
Mac OS X (10.6.8),
iPhone 6 iOS 11.x and an iPad (2)