Tooth whitening at my home is done mainly with hydrogen peroxide.
I had at one time many years ago paid a dentist to make a tray and set me up with a procedure using an expensive gel to lighten and increase the brightness of my teeth.
While the tray for my upper teeth was very useful, I found that the whole thing was too expensive with the cost of visits, and other ways the dentist had to pay for his luxury vacation homes etc.
I felt that I was a victim, and I wanted whiter teeth for all of my life, so I devised a method using the tray with hydrogen peroxide and baking soda.
I use that to brush my teeth with, as well.
You definitely have to rinse afterwards, but the abrasive baking soda and chemical action of the hydrogen peroxide works well together.
Here is a disclaimer:
I am not a healthcare professional so use common sense.
None of this will change the color of enamel caps or tooth fillings or the gold fillings, but normal teeth are whitened this way.