we provide plugins for AcyMailing and Community Builder but until now not for a component like AutoTweet NG.
With AcyMailing you could generate and email newsletters containing your gigs as you like - but that's not what you're searching for.
On my/our site we have AutoTweet NG which can automatically send things to social networks like facebook, twitter, ... (up to two in the free version, unlimited with the paid version) - but this reacts on articles only. I don't know if it is possible to develop a plugin to let AutoTweet react on new/changed JEM events too.
Maybe there are other components providing such functionality and allowing JEM integration, you could take a look at
Social Share category on JED
Personally I'm very busy so I wouldn't like to open this additional task but maybe someone else has time and interest (latter is no problem I'm sure) to develop such a plugin if possible.
Not a solution but hopefully an inspiration...