This only works in special circumstances:
1) the school does not have an event coach in the area; This could be distance, could be hurdles, usually something like the pole vault. Some small private schools may not even have a track program so they train with another school.
2) the original coach is fine with it. Lots of times a coach just wants the most success possible and will sign off on letting someone else coach the athlete; if there is an agreement, it's fine. Believe it or not, good coaches, even ones with big egos, will let kids outside their event area train with someone else.
3) There is a well ironed out plan and understanding: Like, at the state meet, this person may not be able to run only one race, or they do actually have to run the conference meet.
Some of the best athletes in the country were "privately" coached and this was done the right way with ZERO conflict.
Also-- many top programs in the NCAA like New Mexico, Florida State, Oklahoma State, and others have had foreign athletes use private coaches.
In a nutshell, if this is done the wrong way, it is a disaster. It's up to the coach in charge to make the decision and if they do a wishy washy job enforcing what they want, that's when problems arise. Either tell someone "no" or make it work.