Is 3D printing safe for your health?
There is very little information about safety available, as home 3D printers are relatively new. However, plastics such as ABS have a long history in making plastic products, and a study found that at traditional manufacturing methods (such as injection molding and hot wire cutting) do not release dangerous levels of carcinogens and/or respiratory sensitizers in to the air.

Of course, 3D printers are not among the processes covered in the study. In home 3D printing circles, this study that looks at ultrafine particle (UFP) emissions, is often cited. It finds that printing ABS releases relatively high levels of UFP's and PLA releases significantly fewer (but still quite a large amount). However, it is unclear whether/how dangerous these UFP's are in the amounts emitted.

It is often suggested that PLA, partly because of the reduced UFP emissions is safer to print than ABS, partly because of its "natural" origins as it can be derived from materials such as cornstarch. I would caution against this line of reasoning since "natural" materials can still be poisonous (snake venom is natural, after all) and the cornstarch is heavily processed so it hardly resembles its original form. The lower UFP emissions may suggest it is safer, but the study is only quantitative, not qualitative.

That said, PLA does probably pose less of a risk (despite my earlier argumentation against "natural" materials, PLA does play quite nicely with the human body), but I contend the risk with ABS is not too large anyways, given that it has been safely used in factories for decades.

Another study is often miscited, supposedly saying that 3D printing ABS releases hydrogen cyanide. The study only looks at the thermal decomposition of ABS, which happens at significantly higher temperatures than are reached during printing (but a significantly malfunctioning printer might cause toxic gasses to be released, but I contend that at that point you should worry about your printer being on fire, rather than temporary exposure to some toxins).

There are no printers out there that are fundamentally safer than others. However, some printers have an enclosure (containing the fumes) and some even have a carbon filter and a fan for fume extraction. If you would like to err on the side of caution, this might be a good choice (but again, it is not clear if a carbon filter is totally effective).

Finally, as printers are generally quite noisy it tends to be preferrable to keep your printer in a separate room from where you usually work. In this case, fume exposure (during the few minutes that you go to check on your print) is minimal, and the potential advantages of a "safer" printers or using "safer" materials diminish.

Incidental exposure as a hobbyist is probably not a big deal; workers in factories are exposed to the fumes of melted plastic their entire lives and they don't seem to be dropping dead. On the other hand, if you are going to be printing structurally then it is probably preferable to move your printer to a separate room, if not because of health and safety because of the noise.