tl,dr: there is no way for you to access your friend's shared library while he is playing Dota2
An online Steam library can only be in use by any 1 person at a time (in use = any game inside that library is running).
You cannot access someone else's shared libraries while in offline mode yourself.
However, if the owner of the library boots Steam in offline mode, the Steam servers have no way of knowing that he is in fact playing and hence, the owner and 1 other person (so 2 in total) can use his library (note that this is useless in your case since Dota2 is an online game so offline mode is not an option).
Only thing I have not tested since the friendlist overhaul is, whether logging off the friend system is enough to count as offline, or if you still have to completely boot Steam in offline mode (which I suppose).
Apparantly the owner can no longer switch to offline mode in order to share his library while playing, see the comments by Joachim for more details.