I don't blame you for being concerned. I don't ride a lot of miles on MUP's, but I would say that mile for mile I have at least as many close calls on them as on the road. As mentioned above, it's not just other cyclists. I have almost been clotheslined by dog leashes, walkers/runners suddenly changing directions, and vehicles exiting driveways in front of me.
To avoid a serious accident as much as possible:
Make yourself visible with bright clothing
Make noise with a bell, horn, or voice
Anticipate dangers and be prepared to evade
Slow down when approaching crowds or possible dangers
Ride predictably so that others don't have to guess
Analyze every situation on its own merit