OK, so for future reference (since this will probably be useful to other people as well) - I installed dconf-editor (sudo apt-get install dconf-editor), browsed down the registry tree way down into the MATE panel configuration (org > mate > panel > general) and removed all the extraneous entries under the 'object-id-list' key.
So, to begin with, I had something like this stored there:
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['menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'object1', 'object2', 'object3', 'object4', 'clock',
'menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'clock',
'menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'clock',
'menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'clock',
'menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'clock',
'menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'clock']
End result (after removing all the unnecessary stuff):
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['menu-bar', 'notification-area', 'show-desktop', 'window-list', 'workspace-switcher', 'object1', 'object2', 'object3', 'object4', 'clock']
This fixed it for me (no more duplicates on the MATE panels, yay). However, there's a caveat - you might have to do this again once in a while, since after a few reboots I got some duplicates again.
Oh well, at least now there's a way to get rid of them.