Typically, black smoke is related to incomplete fuel burn (carbon exiting the tailpipe). Given that you associated the black smoke with the transmission "going out", it's possible that it was just a fuel dump due to a sudden/surprise "no load" condition on the engine, "free-spin" if you will. You should be able to start it without worry, assuming everything was good with the engine before the transmission stopped functioning.
The only physical connection between the engine and the transmission is the flex-plate/flywheel-torque converter. There's nothing from the transmission that would appear in the exhaust.
As far as what happens when a transmission "goes out", that's pretty broad and "going out" can mean different things.
I've had my transmission lose function due to broken bands/clutches, planetary gears losing teeth, and even the torque converter breaking. In none of those cases have I witnessed what you described as related to the exhaust.
