I really dislike reading replies that start out with "Well, I don't do the thing your asking about, but I'll chime in anyway" but, well... I don't do obedience but I'll chime in anyway.
There are LOTS of people who do both, successfully, without issue. In many ways it is no different than a SchH dog doing therapy work. The dog is quite capable of figuring out what "game" it's playing- whether that be running really fast and jumping over stuff, or staying by my side sitting on command doing the most boring sport ever.
C-ATCH Pimg - DOB: 2/06, UJJ (x2), DJ, HIT, CA, CGC, High in Games, Reserve High in Standard- CPE Nationals 2014
High Jinks vom Neuanfang RATO, CA - DOB 9/12 (Gotchya Day: 1/23/2013) agility superstar in training