You can already do that using the central view. There's an option in the LiteSpeed Console in the top menu under 'View'. You can choose between "Local" or "Central" Instance View, like in the attached screenshot.
So no matter who registers a particular SQL Server with that LS console, any other user later using that same LS Console would have that SQL Server already registered.
There is this on Supportlink:
https://support.quest.com/litespeed-for-sql-server/kb/157410/what-s-new-in-litespeed-for-sql-server-8-1
Centralized Instance Sharing
DBAs can now share instances and associated groupings between clients simply by enabling the Central View from the ribbon (View | Instance View | Central)
Registration changes by one DBA are reflected in other consoles automatically, reducing the amount of time DBAs spend registering, organizing, and removing instances
A private set of instance registrations is still supported using the Local option in the ribbon
For what you are doing with multiple different users wanting to share the same configuration (rather than each user wanting their own separate configuration), you want to be using the "Central" option.
And using the “Central” Instance View, you would avoid the type of problem we had seen on the webex in which one LS Console user had added a specific SQL Server and did some LogShipping configurations to that SQL Server. And then a different user logged into that same LS Console to delete that Log Shipping plan and could not, because that particular user didn’t have the Subscriber SQL Server registered. If both the users do all the configuration using the “Central” Instance View, they won’t have this type of problem
We also have this documentation about the Central View in the online help files too under the section “About Registering and Grouping Server Instances. I have attached a screenshot and also here is the text below:
Centralized instance management:
---------------------------------------------
LiteSpeed has the following capabilities for centralized instance management:
• Shares a common set of instance registrations and maintain grouping across all consoles.
• The Backup Manager View tab is used to switch between local and central repository views (if a central is used).
• The central does not store passwords for SQL Server Authentication. Passwords are still stored locally for the user.
• In the central repository view, all instances are read from the central, including their proper grouping.
• In the central view, any instances that are added, removed, or sorted, automatically get persisted into the central.
....