Hi,
The App stores the pictures (or files) in ~/Library/Messages/Attachments and then strangely numbered folders.
Using the Cover Flow view and displaying the Modified column (Finder > View Menu > Show View Options) can help pin down the date.
The ~/Library is usually hidden on most people 's computers.
To access this Library use Finder window > GO Menu whilst holding down the ALT key then select the Library that appears.
However the app does not Write the info directly to the Folder it uses the route ~/Library/Containers/com.apple.iChat/Data/Library/Messages and the Alias that then points to the Attachments folder.
I am not entirely sure at which point in the process the File Transfer windows say it "received" the file compared to when the app gets it stored (if it can) in the Attachments folder.
I would right click the Attachments folder and then choose Get Info.
At the bottom of the Panel click on your name then Unlock the Padlock.
Then use the cogwheel icon under the list.
You should have Read and Write Permissions.
The text in the Cogwheel icon bit should be greyed out (it reads "Make yourShourName (me) the Owner")
You should also check your Own the Messages folder it is in.
Checking all the folders in the Write path can take a while but should not be dismissed.
8:44 pm Saturday; December 5, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad