
If going in someone elses car the owners dogs should be removed and yours put in first, otherwise your dog may feel too intimidated to get into their space, and the other dog is likely to just be ever so slightly guardy of it's owners car, but in my experience once the other dog is already in there before them then they are fine.
This is something my friends always did when taking someone elses dogs in with theirs, and I have found it works well.
I sometimes travel with a friend whose own dogs get quite guardy and won't let mine in, but after being removed and mine put in first all is well.
They do know each other of course and are on good terms.