Indexes owned by different user to table

I have a schema, within it a load of tables and a load of indexes. This is a managed schema from a COTS product, so I am not happy about adding anything in here.

I have a second schema, this is a bespoke monitoring user, used to gather information about the first schema. in here I am happy to create objects.

If I create an INDEX in the schema MONITOR on a TABLE thats in the schema COTS, (no problems yet)

I expect a SELECT statement run by MONITOR to use the INDEX, but will a SELECT statement run by the user COTS also use the INDEX?