Depends on the contingencies agreed upon during the original sale. Was it used? Defective? Why was it first discounted? The word "sale" is usually synonymous with the word "problem". If it's a manufacturer's defect, by all means it's their responsibility. As cumbersome as it is to ship back & forth (waiting, questioning, hoping...) and all that goes along with that, that is the proper way to have the warranty & repair/replacement handled.
If you chose to take it to your local watchmaker, you should consider any original warranty voided & expect those new charges to come from your pocket, as the dealer/manufacturer no longer has any control over the quality or work performed and the liability for any future repairs or malfunctions have now been passed from his hands to yours.
Personally, I don't put up with defective merchandise--regardless of the source. At the first sign of trouble, I'll immediately be looking for either a return & refund or a return & exchange. From my experience, if it starts having trouble at the beginning--to me that's completely unacceptable--it's only going to get worse with time, and if the trouble is that soon in it's life-span then it's only downhill from there.
However, if it was a watch I was in love with, and I simply couldn't muster the patience to wait for the back & forth shipping, and the repairs were something I could easily afford to do, I would happily bypass the waiting and let the local smith take care of it--who in all honesty is probably much more competent than the factory's assembly-line workers who built it in the first place.