I noticed that if I touch the rear sight on my S&W 620 it wiggles a bit.

Is that normal? It is an adjustable sight.

Maybe I just need to tighten the screw on the right side. Will a regular screwdriver work or do I need to get one of those special gun screwdrivers?

I am actually pretty clueless on how the sight is supposed to be adjusted anyway.

The screw on the right side adjusts windage and will not tighten the blade in the site itself. As far as I know, it is not normal unless the newer guns have a loose blade. Take it to a gunsmith and have them look at it to find out why it is loose.
Don't post the same question in 2 different places on the forum. It uses server space that doesn't need to be used, and you'll annoy the owner of this forum doing so. We'll get to your questions.

Thanks for the reply. Sorry, I meant for this to be in the modern revolver section but somehow I ended up putting it in the retrovolver section.

A tiny amount of wiggle can be normal -- I've seen brand-new S&Ws range from tight as all get-out to a bit of wiggle. Manufacturing tolerances account for this, and by design it is, after all, meant to move.