Do you psychiatrists actually know for sure how psychiatric drugs effect brain chemicals? Or is it just speculation?

Observation & Theory. Drugs prescribed by physicians go through some clinical trials prior to fda approval, but it's also true that the for some drugs the exact mechanisms of action & how they specifically interact in the brain are unknown. It's also true that the theory of chemical imbalance causing some psychiatric disorders, such as decreased serotonin causing anxiety and depression is just that, theory.
Drug research. Drug companies spend millions of $ before a medication can be prescribed. There are 2 phases of drug development before use in humans. The 3rd phase of development requires large scale trials in several different centers with volunteers. A new drug has to be compared to an existing drug and/or placebo (no drug).
Speculation? I'm not a psychiatrist. I treat a lot of patients for depression, anxiety, panic, bipolar and add. We do not speculate as to how meds work. We speculate as much as a pilot speculates how to get the plane in the air and get you were you where supposed to go safely.