Basically, I'm printing a contact sheet of a bunch of images for a project. Around how big should each image on the sheet be to make it clearly visible, while not too big? What size do professionals use on their contact sheets?
True contact sheets were made with the film in contact with the paper, so the images were exactly the size of the negative. With digital you can choose whatever size you want.
If you are using it to select images, then it depends on how similar the images are. If they are very similar and you print them small, it's going to be hard to choose which ones are the best shots.
If you print the images at double the dimensions of a 35mm negative (70mm by 48mm), you should be able to fit 12-16 of them on an A4 (8x10) sheet. I'd think that would be a good size for reviewing.
When I shot with an 8"x 10" view camera on film, my contact prints were one to a page. That's all that would fit!
A square format 2-1/4" x 2-1/4" camera printed from 12 to 16 shots on a roll so that's what was printed.
A 35mm roll of film was typically either 20 or 36 exposures. Ditto 1 roll per sheet of 8"x 10" sheet of paper. Always on glossy finish paper so the contrast helped detail rendition. (A loupe or a magnifying glass also was standard equipment on every desk top and if not there, inside the middle drawer.)
So, What's convenient? If you can get enough detail to see what's important in the shot to select it, it's big enough.
Professionals should be able to be relied on for focus, colour balance, and other technical details. The proof sheet should allow you to see the composition, and the "feel" of the image, enough to select it for critical examination and for markup indicating retouching desired prior to use.
Historically they would be used to select which images you wanted to print larger. Now you can do that on the computer. I think there are many good reasons to print something like a contact sheet, but how big you want the images really depends on what the project is, how the images will be viewed, and also the quality of the media it is printed on.