All programs are required to give residents a written and clinical exam each year. Most (all?) programs use the PRITE for the written exam. The clinical one is more variable but I think most programs do a board style oral exam. There is also a requirement that residents pass 3 clinical skills exams during residency in order to be Board eligible. This exam focuses on exam and history skills. How this is being combined with the requirement for a yearly clinical exam varies among different programs.
The Step 3 is, of course, not a psych exam, but most residents take it in their intern year. It is needed for a permanent license. Some programs will require a resident to pass it by a certain year. If they don't they are dismissed from the program. The written psych boards are now usually taken by most residents in the Spring of your 4th year of residency training. As far as I know it is optional from a training program perspective but you need to pass it if you want to be board certified.