I'm not talking about mindless jamming or noodling, but true improvisation, as in taking a melody started by someone else, and expanding and building on it.

That was one of the biggest pet peeves I had in the band I was in until recently, is that even though they billed themselves as a "jam band" (Grateful Dead cover band), there was just quite simply no true improv being done.  Often times I would try and start a jam by playing some hook/melody/rhythmic pattern on bass, and I would be looked at like I had two heads and the band leader would call out a song and start counting it off.

Well it kind of depended on how drunk I was. Actually back when I was in a band there were times we were the only band sometimes due to other bands no shows and we could just jam for hours. Improvising vocals and everything. I sure wished I had recordings some of the stuff was really good even better than stuff that we wrote down and practiced.

As long as the base melody isn't too complex it's pretty easy. The problem I've found is that most people want to make a riff an improv session... and that kind of doesn't work.

I can, as long as I have a drummer to play with that's intuitive, or I get a good groove going on EZD2. As far as other guitarists and bassists go, it depends on the chemistry, and if they can keep time, and have some intuition.

Sometimes it all clicks, sometimes it doesn't..........

I find it easy but some are too rigid to let it flow and run with it. 

Check out the band String Cheese Incident for jamming and improvisations. Man, they are good and no incidents are ever the same.

https://www.youtube.com/watch?v=hvmt_vNlAME

Yes.

That is my job

nope
when I was in college I lived about 2 blocks away from Berklee
amazing people giging in the bars nightly