No. It's just bone and keratin. That's like asking of your fingernails begin to smell after a while (assuming nothing gets on them).
The first one had flesh on it but I removed that and scrubbed the antlers, and the piece of skull I kept, with neat Dettol and with boiling water. There's nothing left there to rot, therefore no smell.
As for the new ones, there is no flesh left on them to rot, and once I shave off the tip of the antler which has gone slighty rotten because of a break, there will definitely be no smell.

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