System configuration is:
Everything was installed properly and worked fine (node.js, jshint installed through npm). At some point the linter just stopped seeing JS syntax errors.
Except for JSHint, some other linters are installed and work fine - PEP8, CSSLinter. No errors in console regarding broken stuff. Reinstallation of JSHint didn't fix things.
How to diagnose the problem and make the linter work?
After 1 day of searches in Google I've managed to find solution. The page with Global Settings brought some light on the problem: https://sublimelinter.readthedocs.org/en/latest/global_settings.html#syntax-map
The cause of problem was another package, that messed up JavaScript syntax after some update.