When running a double /execute command to run any vanilla command, if the player targeted by the second selector does not have permission to run that command then it will fail and the player will receive the "I'm sorry, but you do not have permission" message.

Adding the following command to a command block will succeed when powered:

/execute @p ~ ~ ~ say Hi

But the next command will fail as described above, with the no permission message being displayed to the player in the second selector.

/execute @p ~ ~ ~ /execute @p ~ ~ ~ say Hi

I do not believe this is expected, and will break any command block oriented world which relies on double /execute.