When you setup a linked server - you also have to define the security context for each connection using the linked server. It sounds to me like you have not done this - which will then only show the system catalogs that are available to public.

To validate, go to the security tab in the linked server properties and see what security has been defined. If you have the item 'Not be made' or 'Be made without using a security context' then you would not see the databases you have access to.

If you check the item 'Be made using the login's current security context' - be aware that this will fail from a client if you do not have Kerberos authentication setup and configured correctly.

You can define a specific security context for all users - and specific users. For example, if you select the last item in the list that will be the default security context. In the upper pane, you can then specify specific credentials to be used for specific logins. For example, you could use a default read only for all connections - and specify for specific logins on that server a mapping to a login on the linked server that has read/write access.