Whatever path you go, indeed be professional. Cold calling is going to probably have the lowest success rate. Instead, work your contacts. Ask friends/family/co-workers if they know any DOs (whether personally or professionally). See if they can break the ice for you (i.e. next time they see them, they mention you would like to shadow, can you contact them, etc).
A doctor who is open to shadowing (but may be hard to get a hold of via cold calling) will probably be more receptive if someone they already know (patient, friend, whatever) vouches for you first.
Social connections work wonders.