It is impossible to run zeal from multiple users at the same time. On startup, an error dialog with the message "Another application instance can be still running, or has crashed. Make sure to start Zeal only once." is shown. If one clicks on the "Retry" button, another error dialog with the message "Zeal is unable to start. Please report the issue providing the details below." is shown. The error message shown when one clicks on the "Show details" button is "QLocalServer::listen: Address in use".
This appears to be caused by the fact the zeal always attempts to use an unix socket named /tmp/ZealLocalServer regardless of user. Hence if one user runs zeal and creates /tmp/ZealLocalServer, other users are unable to run zeal, because /tmp/ZealLocalServer already exists, and they don't have permission to overwrite it.
PS: Additionally, the socket /tmp/ZealLocalServer is not removed if zeal crashes.