This is a fantastic topic for discussion and I hope it gets a lot more replies.
As for me, I do think that the roles of DM and Player help develop one another. I am blessed to be in a gaming group where most of the players are also pretty good DM's so I've gotten to be on both sides of the screen with fair regularity.
I think that playing helps my ability to DM because it reminds me about the sort of things that really make the game fun for the players. I sometimes otherwise get a little too caught up in some of the things that I prefer as a DM. For example, when I DM, I sometimes catch myself spending the lions share of the prep time working on one of the overarching plotlines that the party will probably not catch wind of for quite a while, when what I should be doing is adding the nice little details that really make the world come alive for the players.
I also think that being a player reminds you that somtimes the "obvious" clues the DM leaves for the party to pick up on in order to lead them to the next phase of the adventure are not so "obvious" after all. Basically it reminds me not to be "too clever by half".
And I'm not sure that it qualifies as making me a "better DM" but playing gives me some time off to let me focus my creativity into just one character instead of spreading it across an entire world. That helps me avoid feeling burnt out.
I think that being the DM helps me be a better player in many of the ways that have already been mentioned.
I try to avoid doing things that I think would irritate me if I were DMing. And I think I'm better able to sense where the DM wants the game to go and try and facilitate that with the other players.
DMing has also made me a better min/ma...er, ahem, better at "character design". I get to try out lots of NPC's against the party and I think that gives me a feel of which feats, skills and spells work synergistically together.
I also find that DMing forces a good grasp of all the rules and it makes me better at the tactical aspects of being a player. I more easily see the "best move" that my character could make in terms of drawing fewer AoO's or achieving flanking or cover and I find that my knowledge of what constitutes a free, move equivalent, standard or partial action also results in more efficient play.
On the downside, I have a sometimes forceful personality that lends itself to hogging the spotlight. When I DM, this is a good thing because it keeps the players attention focused on me and the story. But when I play, I sometimes tend to run roughshod over the less outgoing players. All I can say is that I try to stop myself from doing it when I notice and pay attention to when their characters should be taking the lead role.
Anyhow, in short, I think the two skills compliment each other and I look forward to hearing more responses.