Posters have raised some good arguments on both sides of the fence, but in the end, the answer probably depends on your program and your abilities.
As mentioned above, some trainers are very competitive and very interested in how their business is represented to the outside world. They have built a reputation, realized certain achievements, and aspire to be competitive at certain levels. The clients represent the trainer in a sense, or at least the outside world is making certain assumptions about the trainer based on the performance of his clients and the quality of their horses. Trainers that are competitive will want to have a meaningful say about the horses that are brought into their barns. These horses will reflect on the trainer and the trainer will be obliged to train them, ride them, fix them and perhaps one day represent them for sale.
On the other hand, there are programs that are a bit more laid back and the trainer may encourage the client to go shopping on their own and run the best prospects by them for final approval.
Finally, it depends on the client’s experience and abilities. There are more than a few ammies that are very experienced, have been riding for decades and are quite capable of making purchase decisions on their own. IME many of these ammies end up with their own private farms, or would be quite capable of running one; which is a testament to their knowledge and experience.
Meanwhile, there are clients that really shouldn’t shop on their own because they do not yet know, what they don’t know. It isn’t a knock on anyone. We have all been newbies at one point or another.
In summary, I don’t think there is any one single answer. It depends on the trainer’s program and the client’s experience/abilities.