Traditionally, MathOverflow moderators only delete comments that are highly problematic. Even in that case, we copied the entire deleted comment thread to tea.mathoverflow.net. Since migrating to the Stack Exchange network, moderators regularly get requests to cleanup comment threads for reasons that are not highly problematic. For example, users can now flag comments as 'obsolete' or 'too chatty', neither of which is highly problematic. Should there be a change in policy regarding comment cleanup?
I don't think MO should change any policies just because we're part of a network now.
Yes, we are going to get a handful of edits, flags, and otherwise suggestions from SE users which have little to no clue about MO and its community, but that shouldn't really bother anyone.
MO worked out just fine, and even better than fine, and it should keep up its trajectory. At least for the time being.
I could imagine the following:
In my mind a main purpose of copying things to "tea" was to have it readily available if a discussion regarding who said what should arise. If you do not clearly anticpate this happening or the comments are part of an already ongoing disucssion, I would suggest you just delte them without "back-up" on meta, on the grounds that now (and this is new, AFAIK) you could still do so latter (or even undelete them if the need arises). (So the doing so 'just in case' seems not needed anymore.)
Complete conversations should stay (chatty and obsolete or not). [Except I am not really sure how this moving to chat works, so I might revise my opinion regarding for very 'chatty' if some information on this should come up.]
Conversations that are already incomplete (due to parts being deleted) and obsolete should be deleted completely. To give an abstract example of what I have in mind (strike-out means already deleted):
User 1: I do not understand the question. As written this is clearly wrong.
User 2: @User 1, I guess p^2 is meant instead of p also, it is likely assumed p is prime.
OP: @User 2, yes this what I meant. I edit the question. [Edit gets performed.]
In this case I see I no merit at all in preserving the remaining comments, and thus they should be deleted.
I have no strong feelings regarding this, but would rather be careful regarding deletetions, as the risk of upsetting soembody outweighs the benefit of having a 'cleaner' site [this for the chatty and obsolete type]. Also, as opposed to edits, these deletions are not transparent for everybody considerable more restraint seems appropriate.