There are different things you should keep in mind for every situation. Learn what applies to you.
The answer for this group of people is very simple: Medicare pays first.
If your plan has 100 or more employees who are then the private insurance pays first. If, however, there are less than 100 people in the plan, then Medicare pays first.
If the plan covers 20 or more people, then the plan pays first. If there are less than 20, then Medicare pays. There is an exemption for End-Stage Renal Disease where the private plan pays for the first 30 days and Medicare pays after that.
These types of insurance will supersede Medicare and will be responsible for paying first. Also note that Medicaid and TRICARE are never the first payer. Claims can only be submitted to them after private insurance, Medigap, and Medicare have all been exhausted.
If Medicare is not the first payer, you can still use it to pay health care costs. If the billing is done automatically, you will know right away if there is a difference between the invoice and what your private insurance will cover. Sometimes service providers will then bill Medicare for you. If not, you can pay the difference and then submit the receipt to Medicare. Ensure that the private insurance payment is clear so that Medicare can see that they are being used as the second payer, not the first. This only works for original Medicare and you will need to submit the receipt and a 1490S form.
If your medicare benefits are administered by a private insurance company, you will need to submit the invoice to that company.