std::enable_shared_from_this; public vs private

Why is that, what do I miss here?

To make shared_from_this work enable_shared_from_this has to know about shared_ptr which holds the class. In your STL implementation it is weak_ptr, through other implementations are possible. When you inherit privately, then it is not possible to access base class's properties from the outside of your class. Actually it is not even possible to understand that you have inherited from. So make_shared generates usual shared_ptr initialization without setting proper fields in enable_shared_from_this.

Exception is thrown not from make_shared but form shared_from_this because enable_shared_from_this was not initialized properly.

And isn't there a way to make it work for private inheritance, since the 'outside world' of the shared_test class does not need to know that it is enabling shared from this...

No. Outside world has to know that the object has special relations with shared_ptr to work properly with it.

isn't there a way to make it work for private inheritance, since the 'outside world' of the shared_test class does not need to know that it is enabling shared from this

shared_ptr itself is part of the 'outside world'; the shared_ptr constructor needs to be able to access the enable_shared_from_this base class subobject of the shared_test object it points to, in order to initialize the private weak_ptr member of the enable_shared_from_this implementation.