How effective are different types of radar in space over large
distances. Is radar significantly different outside of the atmosphere
The only difference between a radar inside the atmosphere and in space is the lack of air to dim the signal. Instead, a usually small angular size of the target, caused by the large distances.
Would you need to generate a more powerful signal to cover multiple AU
distances? Or would you need a more sensitive type of receiver?
Radar is, after all, just the same as shining a flash-light at something, just with radio waves instead of light. Both increasing the brightness of the flash-light, and the sensitivity of the receiver would naturally help, however, hitting the target accurately over such large distances is difficult, which brings us to the next question:
Would passive radar be less effective? Do signals of opportunity even
bounce off objects in space well enough to register in a passive radar
There is in fact a lot of radio noise in space, some of which is bouncing off other objects, or disturbed by for example clouds of gas. Listening to that noise is a field known as Radio Astronomy, one of the major ways of observing far away objects.
For planetary science, radar has also been used, most notably by the Magellan probe for penetrating the thick atmosphere of Venus to generate usable images of the surface.