Dude,
I actually think that the act of having to hold on to a handle helped my running style a little bit, as my natural tendency is hold my arms up high (imagine trying to pretend you're a chicken) and to flaill my arms around. However, running with a jogger forced me to keep my hands nearer waist height and not wave them all over the place
One of my fondest memories of running with my son was on a longish run when we reached a hill, and he noticed I had stopped answering him.
Him - "Why are you not talking to me Daddy?"
Me- "I'm tired - this hill is steep"
Him - "I'm not tired"
Me - "That's because you're not running..."
Him - "Anyway, I don't think this is a hill, but I'm sure you can do it if you try!"
Me - "Thanks son...."
Him - "Come on Daddy, you can do it..."