Example 1 is correct, because example 2 could be interpreted as men bringing multiple girlfriends, which is not possible. Why introduce ambiguity when the first sentence is accurate? Therefore, use the first sentence.
"Guys are allowed to bring their girlfriend to the event" means guys (all the guys, so each guy) are allowed to bring their girlfriend. Each guy is allowed to bring his girlfriend. Of course, guys are not sharing a single girlfriend, so there is no ambiguity possible in example 1.
However, they will both be understood as "each man is allowed to bring his girlfriend," so it doesn't really matter that much.