Let me expand on @cdeszaq already excellent answer with an opinion of my own.
You should strive to let the people that will perform the tasks do the estimates.
The reasoning behind this is that their experience, which is what is going to help them complete the task, is different from other peoples experience, and thus different people might take a different amount of time to do the same thing.
Rarely will you have a homogeneous enough team that everyone does things the exact same way and at the exact same speed.
Thus, passing a case around between multiple people should probably not be done.
Instead, let a parent-case be without an estimate, split the case up into chunks, that you then dole out to different people, and let them estimate their own case. This way, each person estimates only what he/she will be doing, and the parent case gets a good nice total estimate from the sub-cases. Then you don't need to pass anything around at all.
Now, of course, if you actually need to pass a case on to someone else, for instance because someone called in sick and you can't wait for them to return to work, then you pass it on, but that's a different issue.
If you simply want to pass a case around because the work really needs to be split up and doled out to different people, use sub-cases.