Which STDs do not necessarily have any symptoms?

someone sounds paranoid…lots of them, including HIV in most cases.

Basically all of them. Chlamydia and Gonorrhea may not cause any symptoms but still can be transmitted. Syphilis has a long latent phase when it's still transmittable. Many strains of HPV don't cause any visible signs but still can lead to cancer of the cervix, anus, or penis. HIV has a long preclinical phase before it starts affecting the immune system. Herpes infections can occur without any symptoms, or you may have an outbreak followed by no more symptoms for years if ever. Scabies and pubic lice are about the only ones with consistent symptoms.

Point is, if you've been sexually active, you owe it to yourself and your partner(s) to get tested regularly.

Chlamydia- up to 25% of infected men may have no symptoms.
Approximately 70% of women will not have symptoms due to chlamydia
Trichmoniasis- the infection usually does not cause symptoms in men and goes away on its own in a few weeks.
Please see the web pages for more details on Chlamydia and Trichomoniasis.