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Each operator has its own considerations. For example, the << operator (when used for stream output, not bit shifting) gets an ostream as its first parameter, so it can't be a member of your class. If you're implementing the addition operator, you'll probably want to benefit from automatic type conversions on both sides, therefore you'll go with a non-member as well, etc...
As for allowing specialization through inheritance, a common pattern is to implement a non-member operator in terms of a virtual member function (e.g. operator<< calls a virtual function print() on the object being passed).
If you plan on implementing streaming operators (<< and >>) then they will be non-members methods because your object is on the left of the operator.
If you plan on implementing ->, () or [] they are naturally member methods.
For the others (comparison and mathematical) you should check out Boost.Operators, it really helps.
For example, if you want to implement the following operators:
MyClass& MyClass::operator+=(int);
MyClass operator+(const MyClass&, int);
MyClass operator+(int, const MyClass&);
You only have to write:
class MyClass: boost::operator::addable<MyClass,int> // no need for public there
{
public:
MyClass& operator+=(int);
private:
};
The 2 operator+ will be automatically generated as non-members which will let you benefit from automatic conversions. And they will be implemented efficiently in term of operator+= so you write code only once.
For binary operators, one limitation of member functions is that the left object must be of your class type. This can limit using the operator symmetrically.
Consider a simple string class:
class str
{
public:
str(const char *);
str(const str &other);
};
If you implement operator+ as a member function, while str("1") + "2" will compile, "1" + str("2") will not compile.
But if you implement operator+ as a non-member function, then both of those statements will be legal.