If you have worked with me or read my articles you probably know my answer... "It Depends". There are differing levels of trainer involvement for each one of us so there are different considerations for their involvement in your saddle selection.
If you take lessons and very occasionally the trainer hops on your horse to illustrate a technique or make a correction, they should always be asked to view you in a perspective saddle in order to confirm that it places you in a correct position. They should also have a long enough history with you to establish that your riding is as good, or better in this saddle than in your previous one. If your saddle fitter has determined that the saddle is a good fit for you and your horse, and the trainer gives her blessing based on a visual evaluation, you are good to go!
If both you and your trainer are riding your horse on a regular basis, determine who your new saddle is really for. I have some clients whose taste in saddles and whose builds differ enough from their trainers' that they bite the bullet and purchase two saddles. If this is not in your cards, work with your saddle fitter to identify separate saddles that will accommodate you and your trainer and that work well for your horse. An alternative is to ask your saddle fitter to guide you through the fitting process that will best accommodate you, your trainer, and the horse with one saddle. There are no hard and fast rules but the saddle fitter will be able to point out the impact on the horse of, say, a saddle that is too small versus too large in the seat for a rider. My bottom line as a fitter is always the horse but if the saddle is unsuited for the rider, the horse will feel the discomfort also.
Finally there are the owners whose horses are in full training. If this is going to be for an extended period, consider purchasing a saddle that fits the trainer then trading it in for one that fits you when the time comes for your partnership with your horse. I am a firm believer in the used saddle market and encourage clients to buy and sell as their needs change.By the time your horse has matured with professional training, odd are that he or she will need a new saddle anyway.
Some trainers have saddles that can be used on a variety of horses. That said, trainers are not professional saddle fitters and I have seen a great deal of damage done to performance horses by trainers' saddles. It never hurts to have your saddle fitter check any saddle that will be used on your horse. Trust me, it is WAY cheaper than having the hocks and stifles injected because your horse is compensating for pain in the back.
One caveat to observe when looking for a saddle: keep an open mind! Some professional trainers have saddle preferences which work well for them and sometimes they are sponsored or have deals with saddle companies. Do not let their preferences or prejudices in saddle choices affect your selection process. Saddle companies make saddles in a way that will suit certain riders and certain horses. We, and our horses, are all built differently and we react differently to a saddle. Work with an independent saddle fitter who knows the quality and characteristics of many saddles and who can steer you toward the best for your unique needs.