It appears that the TightVNC server package doesn't have this support. At least the version I have (
tightvncserver version 1.3.9-9+deb10u1, from Debian 10) doesn't.
It's also worth noting that the official TightVNC web site says that this version (1.3) is outdated and not supported. Unfortunately, the current version is no longer open source and requires a commercial license. See also https://www.tightvnc.com/licensing-server-x11.php.
As an alternative, I would suggest using the TigerVNC server, which is included in my Debian installation and is probably available for many other distros as well. Using it, I was able to get clipboard sharing working (via the previously mentioned
If you do upgrade to TigerVNC, however, please note the following issue:
The default installation of TigerVNC (at least for me) only allows connections from
localhost. I assume this is for use with SSH-tunneled connections. A one-line configuration setting in either
~/.vnc/vnc.conf or the command-line when starting the VNC server will disable this, allowing connections from over the network.