Generally folks who do group lessons are not going to be offended if you are taking privates, or leasing a horse off site. However, once you have chosen a discipline and a barn, it is take it or leave it. A good competition barn is going to have an all encompassing program that you are either in or out of.
If you are looking to be competitive, you will be spending lots of money on coaching. And it gets worse as you ride higher levels - people at the Olympics still have coaches!!
Things I would say to look for in competition barns - look at the competition results of the high level trainers, and their coaches. Look at the competition results of the high level students. If the highest results at the barn are in low hunters, first level dressage, and pre training eventing, then it's probably not a very competitive barn, and if it is, it's not one which encourages students to push their limits and move up the levels. When reading descriptors, ignore the word "champion" - a horse/rider can be champion of a division that they are the only ones entered in!! Look for high end shows in the area, high level classes, good scores and pinnings out of how many horses. And lastly, look at the treatment of the horses - do they stand in stalls all day? Are they too fat? Too thin? Fit for the work? Happy?
But before the point of getting competitive, it is essential to develop a good seat and choose a discipline. Most competitive barns require you have your own horse, and before that point you need to be pretty sure of what you want to do!