Query against sys.schemas and sys.synonyms runs very slow for one user

You may want to re-write your query as follows (I'm using dbo rather than XXXX so that I do find some synonyms on my testing database). This is similar to the re-write you found to be more efficient, but avoids the need to declare a variable and use two queries.

SELECT name, base_object_name 
FROM sys.synonyms 
WHERE schema_id = SCHEMA_ID(N'dbo')
ORDER BY name

This yields a plan like the following:

One very interesting thing about the Filter operator in this plan is that it has a predicate that performs an internal has_access() check. This filter removes any objects that the current account does not have sufficient permissions to see. However, this check is short-circuited (i.e., completes much more quickly) if you are a member of the db_owner role, which may explain the performance differences you are seeing.

Here is the query plan for your original query. Notice that all synonyms on the database (1,126 in my case, but likely many more in your case) pass through the very expensive has_access() filter, even though only 2 synonyms match the schema. By using the simplified query above, we can ensure that has_access() is only invoked for the synonyms that match your query rather than for all synonyms in the database.

Using sys.dm_exec_query_profiles to explore further

As Martin suggests, we can confirm that the has_access() check is a significant bottleneck by using sys.dm_exec_query_profiles on SQL Server 2014+. If I run the following query using a db_owner account on a database with ~700K objects, the query takes ~500ms:

SELECT COUNT(*)
FROM sys.objects

When run with an account that is not a db_owner, this same query takes about eight minutes! Running with actual plan on and using a p_queryProgress procedure that I wrote to help parse sys.dm_exec_query_profiles output more easily, we can see that almost all of the processing time is spent on the Filter operator that is performing the has_access() check: