I realize this is an older thread but I was having this problem and pulling my hair out. I finally talked to Apple Support who escalated me to a senior advisor. She said this is "an emerging problem" with Big Sur and asked me to send a system diagnostics report. After hanging up she emailed the following suggestions to me. After following the suggestions and restarting the computer, my problem was resolved. Posting this for others that may be having the same problem.
Here are the suggestions from the Apple Senior Advisor:
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Follow the steps below to further test the issue that we are having with iMessages.
Remove orphaned registration keys
Open Utilities folder (open finder>GO>Utilities) open Keychain Access.
From the Keychains list, select login.
Under Category, select All Items.
Search for any "com.apple.facetime: registrationV1" entries and remove them.
Restart the Mac.
Sign back in to iMessage if needed and test for the behavior once more.
If the behavior continues, follow these next steps I provided below.
Remove potentially bad keys in Keychain Access
Open Keychain Access again
From the Keychains list, select login.
Under Category, select All Items.
In the Search field, enter "IDS", then click the Kind column to sort by kind.
Delete all of the IDS items that are of the kind "application password."
In the Search field, enter ”FaceTime”, then click the Kind column to sort by kind.
Delete all found items that are of the kind ”application password.”
Search for ”iMessage” and delete all items with the name "iMessage Signing Key" or "iMessage Encryption Key."
Restart the Mac and test issue once more.