First things first with regards to some of the replies from those living in the New England area (I'm originally from the North Shore region of Massachusetts). Rotaries and even the mostly-long-gone traffic circles that used to be all over New Jersey typically have a larger diameter than the roundabouts that have recently been planned/built in the US. That said & having grown up with rotaries; the general premise was to use turn signals after one's entered the rotary to either change lanes within or to exit NOT prior to entering. The reason being is that the entrances to the rotaries/circles are essentially merge-style ramps that only allow for one-way movement.
HOWEVER & obviously, if the related rotary/circle and has entrance ramps with more than one lane and one wishes to change lanes (where allowed) prior to entering the circle; then, yes one must use their turn-signal.
It's worth noting that several existing rotaries in Massachusetts have recently been restriped to multi-lane roundabout specs per the MUTCD. The noticeable change is increased advance signage regarding which lane to be in prior to entering the rotary with regards to route & destination and more channelized lane striping upon entering & exiting the rotary. In some instances, the exit ramps from these rotaries are striped as either two lanes or two lanes spaced further apart via cross-hatched striping. Such separates traffic that just entered the rotary wanting to exit at the next immediate ramp to simply from the inside-rotary traffic.
Regarding recent smaller roundabouts that have been built in the Bay State; such seems to operate in a similar manner as the legacy rotary but on a smaller size & scale. With these & per the original question; one only needs to signal while in or to exit the roundabout but not (except for lane changes if any) upon entering since it's is a merge-style ramp.
Signaling your exit from a roundabout is required in many states, though some states have more general rules about 'signaling your path'. Signaling left until you're ready to exit will also help motorists not jump in front as you go around (etiquette). I think the confusion stems from rules about when to signal and distance in advance of an intersection. I've read law enforcement opinions that depend on each leg of a modern roundabout being a separate intersection, and that is incorrect. The roundabout is a single intersection and the roundabout is part of the traffic control for that intersection. Just like any other intersection, drivers should signal their intended path in advance of arrival, right or left, with the added rule about signaling your exit from the circular roadway that many states have specifically adopted.At the 20 mph operating speed of a roundabout, signalling is quite easy. I use my signal in advance of entry if turning right or left, and just past the splitter island just before my exit.