Training at other gyms in BJJ is a tricky situation. This is because of the dual nature of BJJ. It is a martial art, but it is also a competitive team sport at a lot of places as well.
The goal of any martial art is individual improvement. No matter what level you start at, you want to become the absolute best you can be personally. The only real measurement of your progress is your own goals, so how you are progressing compared to others is not that big of a deal.
From a martial arts perspective, ego free training at other schools is perfectly fine. In fact, it ought to be encouraged. It raises your own level to be exposed to different styles, and it also simultaneously raises the level of your new training partners as well. In the martial arts, this is a good thing.
However, BJJ is also a competitive team sport. The goal of a competitive team sport is not to be the absolute best you can be. The goal is to defeat your opponents. This is where the complication of training with other teams arises. Mutual benefit is not the goal (at least not a direct one). In fact, giving any benefit to a competitor is discouraged. This is simply the nature of competition.
This isn't just true of BJJ either. Any competitive martial art will have this issue. There were some posts on here about how Judo is so much more open about training. I am sure those stories are true, but it's only because competition must be stressed less than the martial art aspect at these schools. If you think about a Judo Olympic team, there is no way in hell that one guy from one team could just pop in to practice with a rival team right before the Olympics. The coach from one team will not train competitors from the other. That would be a conflict of interest from a competitive standpoint.
These kind of training restrictions are one of the main drawbacks to competitive martial arts. Of course, there is also a lot of clear benefit to having competition in the martial arts. I personally like the competition aspect. But there are some negatives which is why even some really practical and effective arts swear off competition.
The dual nature of martial art and sport in BJJ is where the trickiness and variation comes into play. If we were talking about football and not BJJ, it would be very obvious as to why you can't just pop in to practice with a rival team. However, since BJJ is much more than just a competitive sport, it gets complicated.
I've seen BJJ schools run the gamut from completely cool with people training other places to expelling anyone who ever does so. I don't think it really has anything to do with Brazilian versus American, traditional versus progressive, etc. Mostly what I have seen is that the schools that have the strictest regulations about training with other gyms also have the most serious competition teams.
It's not necessarily right or wrong; I think it just mostly comes down to whether you view BJJ as a martial art or a competitive sport.