If the table is not in the default schema of the user, or not in the dbo schema, you have to specify the schema. For some type of objects - UDFs - you must specify the schema regardless.
What I suspect may be happening to you is that the default schema for the users may be something different from dbo. In lot of cases it is useful and you do want that. However, especially in the case of windows authenticated users, the default schema becomes the same as the username. Changing their default schema to dbo may be what are really looking for. You can change a user's default schema in the properties dialog under the DatabaseName->Security -> Users in the SSMS Object Explorer.
Changing the default schema would affect only objects created going forward. You can change the schema of existing objects if you need to using
-- in this example, YourDomain\YourUser is a windows authenticated user
ALTER SCHEMA dboTRANSFER [YourDomain\YourUser].Tablename;