Pass by value or const reference?
In this particular case, and assuming C++11 and move construction/assignment for strings, you should take the argument by value and move it to the member for the constructor.
A::A(string str) : mStr(std::move(str)) {}
The case of the setter is a bit trickier and I am not sure whether you really want/need to optimize every bit of it... If you want to optimize the most you could provide two overloads, one taking an rvalue reference and another taking a const lvalue reference. At any rate, the const lvalue reference is probably a good enough approach:
void A::setString(string const& str) { mStr = str; }
Why the difference?
In the case of the constructor, the member is not yet built, so it is going to need to allocate memory. You can move that memory allocation (and actual copying of the data, but that is the leaser cost) to the interface, so that if the caller has a temporary it can be forwarded without an additional memory allocation.
In the case of assignment the things are a bit more complicated. If the current size of the string is large enough to hold the new value, then no allocation is required, but if the string is not large enough, then it will need to reallocate. If the allocation is moved to the interface (by-value argument), it will be executed always even when it is unnecessary. If the allocation is done inside the function (const reference argument) then for a small set of cases (those where the argument is a temporary that is larger then the current buffer) an allocation that could otherwise have been avoided would be done.
The article you site is not a good reference for software engineering. (It is also likely out of date, given that it talks about move semantics and is dated from 2003.)
The general rule is simple: pass class types by const reference, and other types by value. There are explicit exceptions: in keeping with the conventions of the standard library, it is also usual to pass iterators and functional objects by value.
Anything else is optimization, and shouldn't be undertaken until the profiler says you have to.