I am impressed by the mechanism that allows the PT111 to have trigger function in both modes, and I would say the DA isn't all that heavy, while the SA isn't the most crisp break I've felt.
If you find one to fondle, try to get a field-strip to look at the internal hardware.
Remember, most SA/DA pistols with hammers will let you drop the hammer on a round as many times as you like, the Taurus just does it with a striker and a traveling sear (I don't know what else to call the gadget that cocks the striker and then drops to release it, anyway)
That being said, with a 12-round magazine, the "second strike" feature is nice, but tap/rack/target drill had better follow after that second trigger pull, because I'd bet that you're pulling the trigger on an empty chamber more often than you are pulling it on a hard primer. I've seen hard primers on centerfire a few times, and a second firing pin hit has worked about 50%, but that's at the range with plenty of time to eject/assess/reload the bad round after waiting for a hangfire.
If you must have the feature, Taurus is coming out with a slim model, I don't know if it will have the second strike gadgetry built in, but if it does, I believe it is intended to compete with the PPS/Kahr/Kel-Tec mini nines, and might be worth the wait.