I am a trainer, and I usually don't get anything from the owners except occasionally a halter that the horse was wearing when it arrived.
IMHO, it would just be too difficult to keep up with all the stuff if owners shipped all kinds of tack with their horse off to training. Just for me, I've got 5 saddles, about 25 halters, and well over 30 bridle/bit combinations, not counting the different pads I use to modify saddle fit if I end up with a horse that doesn't fit my saddle just right.
A good trainer will have just about everything they need to fit just about any horse...or the means to get what they need without borrowing from the owner of the horse.
Of course, it's something entirely different if you have some tack that you want to use and the trainer doesn't have it. I've done that several times, where an owner will show up to pick up their horse with entirely new tack; saddle, pad, bridle, bit, etc. In that instance, I will take all the tack, make sure it's adjusted and fits properly, then I will put the first ride on the horse in the new tack just to ensure that the difference isn't going to surprise the horse into misbehavior.