The latest standards from AERA, APA, NCME require test publishers (which includes surveys, self-report tools, etc.) to collect evidence to support the interpretations they claim can be made of the test results. That doesn't mean they all do, and instruments developed by researchers for their own research usually lack that evidence. Whether or not a test has such evidence largely determines its quality. Higher-end (expensive) tests like Student Self-concept Scale will pay for the research to support it.
The whole subfield of supporting certain interpretations of test results is called "test validity," which is slightly different from either logical validity or scientific validity. The popular model is based on the work of Lee Cronbach, but the most advanced model (which is canonized in the latest standards) came from the work of Samuel Messick. The Wikipedia articles reflect this duality with "Validity (Statistics)" describing Cronbach's view, and Test Validity describing Messick's.
To answer your question, correlation has been an enormous part of validity, to the point that a correlation coefficient has been called a "validity coefficient," though this terminology is falling out of favor. (As a graduate student, I was humbled by an established leader in the field when he dismissed my correlations with, "You can get anything to correlate.") Correlation is an important tool, but it's a first step.
Some studies do ask other people to verify someone's self-rating, and some scales (e.g. The Vineland Adaptive Behavior Scale) have others (informants) fill out ratings on the examinee. The examinee never even sees the test (though the examiner must have their or their legal guardian's permission).