Are butt implants safe and do they feel natural?

If performed by a buttock implant specialist, buttock implant are no more dangerous than fat transfer or breast inplants:
Risks and potential complications of buttock implants are very similar to breast implants. One main difference is while breast implants have a nearly 20% risk of hardening from excessive scar tissue formation surrounding the implant (aka capsular contracture), in my experience buttock implants have ~2% risk of this. 

On the other hand , because at least 80+ % of the fat transferred will melt away within a year, most patients are not good candidates because they lack an adequate amount of fat to harvest. Another tip is that if you purposely gain weight (i.e. fat) for the procedure, the fat you lose first as you get back to your baseline weight is that very same fat that was transferred into your buttock...so don't fall victim to this recommendation. Even those patients that had adequate amounts of fat pre-operatively, still end up seeking buttock implants after a year or so because most of the fat transferred melted away leaving them with minimal result.

 Although overall using your own fat is relatively safe, it not infrequently melts away unevenly leaving one butt cheek bigger than the other or with dimpling or hard fat cysts.  The one serious complication that can rarely (< 1%) happen is "fat embolism" in which some of the fat gets into the blood stream and travels up into the lungs, heart, and/or brain causing serious problems. This complication is more likely to happen with the larger amount of fat being transferred. This is even more likely to happen when using fillers like liquid silicone, PMMA, Sculptra, and hyaluronic acids. Also fillers, when injected in large quantities, have a relatively high infection rate, guaranteed tendency to migrate away from the original area they were injected, and almost always stimulate a lot of inflammation with a subsequent and disastrous amount of scar tissue/hardening.
Thus buttock/hip implants become a very good, safe, and long term reliable option for most patients seeking buttock/hip augmentation (at least in my practice).
Although it takes 6 - 7 months for buttock implants to settle into their final position and fullness, it can take a full year for them to completely soften. This is how long it takes for all the micro-inflammation to disappear. I tell patients, in the end, they will feel similar to an Olympian sprinter's athletic buttock. If you surf around RealSelf, you will see patients twerking with buttock implants...so yes they can really "jiggle" as well. 

Butt implants are placed inside the muscle of the buttocks and although they are firm, when in the muscle they are cushioned by the muscle; regardless, the feel of the buttocks is much like a dancer's buttocks, strong and firm and not like a fatty bottom feel, one which you get with a BBL.