Keep in mind interface does NOT contain any implementation data. When you add property in an interface, it merely says that a class implementing this interface needs to have said property with get, set or both methods, depending on what you wrote. So any class implementing your interface has to implement (or have auto-generated) p1 property with get and set method, p2 with get method, and p3 with set method. Interface doesn't care whether these will be auto-generated or your own custom implementations, they just have to be in an implementing class.
Therefore, you can write
int p { get; }
in an interface as all it does is telling that any class implementing this interface has to have property p with getter, again, not caring about its actual implementation - you could write a getter that does some computations, returns some constant, etc. OTOH in a class writing the same would mean that you want a property with auto-generated backing field, except since it would have no setter, you couldn't actually change its value, so it would always have its default value 0.
And as noted, you cannot write access modifiers in an interface, as all interface members are implicitly public.