Don't challenge or dare a person to commit suicide thinking you will shock them out of the idea. Do not worry about invading someone's privacy even though they try to get you to promise secrecy. This is not a test of friendship but a response to a deadly illness. Don't leave it up to them to get help on their own. Try to arrange for professional evaluation and treatment.
Some people experience suicidal impulses even though they are under treatment.
The patient needs to have their treatment plan adjusted. Some people have no control over the death wish that sweeps over them, yet they understand their impulse is irrational. They want people to intervene. If a chronically suicidal person lives alone, arrange for friends to check- in with the patient by phone every day. Ask for the phone number of their treatment doctor and which hospital accepts their insurance should a crisis occur.