As I understand it you have two users SCOTT and SCOTT1 in DB02 and SCOTT can get access to data in DB01 via a public database link but SCOTT1 cannot.

There are two main possibilities and a minor possibility and it would be useful to know what error message SCOTT1 gets to resolve which of the possibilities applies.

For the PUBLIC database link to work with SCOTT and not SCOTT1, SCOTT must be a user in both databases with the same password in both databases and grants on the tables.

One possibility is that SCOTT1 does not exist in both databases or does not have the same password in both locations. In this case he should get ORA-01017 (followed by ORA-02063).

A second possiblity is that SCOTT1 exists in both databases and has the same password in both locations but does not have the required privilege on the table. In that case they would get error ORA-00942 (again followed by ORA-02063).

A less likely possibility (but still a possibility) is that one or the other of these users is actually using a private database link with different properties. In that case it could be the link rather than the users that explains the differences in access.