I have two tables
You could introduce a
PERSON-table. This is
1:1 related to your doctors and patients (and later maybe to employees, suppliers, institutions, whatever). Let the contacts be related to this person-table.
Application code can model this with inheritance quite easily...
Within your person-table you keep some general information like
PersonType (reference to a person-type-table with entries like
Patient and ...).
Keep this table slim...
If you have to choose out of your own ideas, I'd prefer the second. Keep them in one table and mark them with a type column. Avoid speaking keys...