Lots of great answers here. Anything I would have chimed in with has been pointed out by someone else. I've played guitar for 16 years and did not dare to sing a note in front of anyone until about a year ago. As you may guess, it was absolutely terrifying.
I was in a band with a great singer for 5 years, and when our band broke up we decided to stick together and do our own thing. Now I've taught him to play rhythm acoustic and he taught me a couple things about singing.

A lot of people have said it already, but it's really all about confidence. We are all animals after all. If you're terrified and you get up at open mic, the audience can smell that fear. If the people watching have a doubt in their mind about how good you are, it's because YOU put it there. In my opinion, a good player/singer is just someone who practiced properly and knows how to be a confident salesman in front of a crowd.

Another thing to consider is that if you're struggling to play and sing at the same time, play to a metronome and start by making sure you change chords in the right spot. If theres one thing i've learned since I started singing while playing, it's that once you open your mouth and words come out, no one cares what your hands are doing as long as the chords change when they should.

I'm a hardcore perfectionist about nailing guitar parts, but learning to sing and actually singing lead with a band has taught me so much that I wish i'd have realized years ago.

I think everyone in a band should spend time at least tinkering with every instrument in the band. If we all understand each others role more clearly, then it's way easier to stop waiting your turn and just make good music instead.

if anyone wants to check it out, heres a song i wrote for my wife, one of my first stabs at singing lead for the band:

www.soundcloud.com/the-stones-throw/shots-fired