I log quite a bit with a roll-my-own logging system. But I all have database errors sent to my phone using PushBullet. I don't put the details of the error in the notification since I'm in healthcare and can't expose certain information. So, the error tells me which client had the problem and a stack trace. From there, I just log in and look at the logs where the real error exists.
I considered using dweet.io at one point, but it's too public for my taste. Cool concept, nonetheless.