Not usually, no.
A flare returning to its owner's hand after use is a standard rule for the card type, and not specifically one of the "effects" of each flare; thus, Wild Mimic does not steal the flare it is duplicating. On the other hand, if a particular flare says to give it to someone else after use, this is an effect of that flare. In most cases this distinction will be moot (for example, Wild Disease is given to Disease or discarded, and then your duplication redundantly causes it to be given or discarded to the same person or discard pile). However, since Wild Reincarnator and Wild Changeling are given to whichever player the flare affected, then your duplicate use of the card can indeed change who ends up holding it.