As someone who has done a fair amount of 1x1s himself, it's always been a post-to-post basis. Sometimes my posts have exceeded past 1k words and sometimes I barely passed a 300-word marker. Sometimes my posts were a single paragraph (not counting dialogue) and sometimes I wrote over 10. It all depends if what you're writing is a reactionary post or if you're setting the stage for some events. 

For 1x1s, I always felt that each person controls a certain portion of events. So in short: they do as much as you feel they need to. Just make sure you and your partner understand this and all will be smooth like butter.