The second that associations required us to check equipment they gave us permission to enter a dugout. If it's that big of a deal, have an assistance coach stand by when you check it. Always check it with your partner.

That is not a logical assumption.
Why can't the association have more than one requirement in regards to cheking equipment?
The association requires the equipment to be placed outside the dugout and the association requires the umpires to check the equipment.

Did they also give us permission to check the bat bags that are in the dugout? What about other personal bags that may be in the dugout that a bat or helmet can fit in?

No thanks.

I think you are bright enough to see the difference between looking at a bat leaning up against the fence and unzipping a player's bag. Or at least I hope so. If that's the case, your comparison is without any merit whatsoever.

We have not been told one way or another about entering the dugouts. The association I am a member of does not have a rule regarding entering a dugout or not entering. The only rule was do have is that when entering a dugout, we are told to have both enter the dugout together and both leave together. When we are doing equipment checks normally nobody is in the dugout anyway, except maybe a coach doing the lineup card.

In my personal opinion, the biggest issue is that some areas may have had items stolen and the only time non-team members were in the dugout was when the umpires checked equipment. Even if the umpires had nothing to do with disappearing stuff, they were an easy target.

It does make it easier when the coaches have players bring out the equipment to be checked outside the dugout.

Luckily the middle schools I umpire don't have dugouts, only open air benches, so when I am doing MS 1 man fast pitch, I don't have to worry about going into a dugout.