I was in a witch cave shooting fireballs at skeever and one ran up and attacked my companion before he killed it. It got me thinking, can they get diseases too?
I can't see any way of knowing though.

No idea, but it can't hurt to make them use a shrine now and again, just to be sure.

I've seen them get that khaki glow that says you got a disease, but being that I play on a console, I can't actually look at their stats to tell if they got one.

/All I know is my gut says maybe.

I don't think followers can catch diseases, if they could there would have to be some way set up for you to know it and address it, in the same way that you can see their health bar if their health is diminishing and heal them or give them potions. I used to give followers all kinds of potions for Curing Disease, Restoring Stamina/Magicka, etc. The only potions they ever seem to use are the Health ones. So my assumption is if there's no way to address an issue like that with a follower, then the issue simply doesn't exist for them.

This is an interesting question. I know I have never needed to cure my follower, but I never checked for diseases either. I figured they would complain or something.

I know they use potions, so what you could do is give them a Cure Disease potion, then see if they ever use it. Note, these have to be stock potions, not ones you brew up.

I looked through the Creation Kit to see if I could find a definitive answer to this question. I could not find any scripting on this. Diseases are handled as spells which means a spell is added to the target. Adding spells to a character in game requires the ref id of the character. It is unlikely that when a vampire, for example, attacks Lydia that the game is set to find the id number and add the spell (Sanguinare Vampiris). The player, on the other hand, has a specific id and commands are available which target the player specificallly.

No they cannot catch our contact diseases