"A CREATE command specifying an unqualified object name creates the object in the current schema (the one at the front of the search path, which can be determined with the function current_schema)"
For example:
Is there a way to run the create/insert statements below without prefixing the schema name, user1?
As the user, user1:
Create table user1.table1 (id int);
Insert into user1.table1 values (1);
Just omitting "user1" and seeing what happens would be informative. You should find it does exactly what you think - namely because the default search_path will cause "user1" to appear first.
Insert is more similar to Select than it is to Create - the object being inserted into must already exist
