It depends. It benefits the gym to have better coaches but the gym should have the choice of what training/who gets the training, I suppose. I wouldn't assume the gym should pay for whatever training I might feel like getting. If the gym wants me to upgrade my training because they want me to move into other areas, then they should pay for it. It will still benefit both the gym and the coach (gym will have better coach, coach will have better education). I think also it could be a matter of longevity. I think a gym would be more likely to pay for further education for someone they have every reason to believe is there for the long haul vs someone they think will probably leave in a year or so (college etc)...
ETA: for training camps that coaches accompany gymnasts to, it's partially the gym and partially the booster club here (such as the Texas TOPs camps etc). It does benefit both the gym and coach still, and obviously the gymnasts cannot go alone. The coach(es) should definitely not be out any personal costs. Hope that made sense!