Brian,
The answers to your questions depend on what you wish the stern anchor ultimately to do, as some of the respondents have noted. The critical situation would be if anchored relatively close to shore for say, shelter. So the shore is windward of you. You wish to put out a stern anchor to hold the boat offshore (i.e., does not allow it to swing around and drift toward shore) if there is a change of wind direction or other conditions. So, the stern anchor, if that occurred, might be holding the boat against the wind and sea. It should have the same holding power as the bow anchor. In my opinion, if anchoring overnight in that kind of setting, bow and stern should have the same holding capacities. That is, of anchor sizes rated for the vessel, with the same strength of rode, & SET to similar scopes, etc.
It's a different situation if the boat is anchored bow toward sea and a stern anchor is wished to keep the vessel aligned bow-on to the in coming surge or swells; or to prevent the boat from yawing extensively, as you noted. In that case, the stern anchor would not need the same holding power. So, it might be a smaller anchor set on reduced scope, etc., under the expectation of no significant change in conditions.