Cross Database Ownership chaining, by default, is disabled on SQL Server. This means that if 2 objects in different databases have the same owner then permissions will not be implicit.
Let's say you have a stored procedure, called proc1_sp and a table, called table1_tbl in a single database (DB1). Both proc1_sp and table1_tbl are owned by dbo. You have a Login called TestUser, which also has a user (TestUser) on DB1. Within that database you have granted TestUser permissions to execute proc1_sp, and that's it (no db_reader role for example). The SQL for proc1_sp might be something like:
ALTER PROC proc1_sp @id int AS
FROM Table1_tbl T1
WHERE T1.ID = @ID;
If tsetuser were to run this procedure, then the data would be returned without a problem, as the permissions are implicit, as they have the same owner. Say, isntead, however, you had another table Table2_tbl with was owned by Steve. If you had the Same SP, but replaced Table1_tbl; with Table2_tbl then the SP will fail, as they don't have permission to select from the table.
The same is true for cross database queries, apart from even if the owner is the same, the implicit permission is lost cross database. If you want to enable the user to be able to SELECT from table2_tbl you'll need to also grant that permission. If that able is on a separate database, then a user linked to their login will be needed.
You can, like discussed above, enable Cross-Database Access, Enabling Cross-Database Access in SQL Server. That does come with caveat's, howerver, such as:
Database owners and members of the
db_ddladmin or the
db_owners database roles can create objects that are owned by other users. These objects can potentially target objects in other databases. This means that if you enable cross-database ownership chaining, you must fully trust these users with data in all databases.
Users with CREATE DATABASE permission can create new databases and attach existing databases. If cross-database ownership chaining is enabled, these users can access objects in other databases that they might not have privileges in from the newly created or attached databases that they create.
Excuse my typos and sometimes awful grammar. My fingers work faster than my brain does.