How are witness operators physically implemented?

Let's take an example of an entanglement witness of the form $$W = | \phi \rangle \langle \phi | ^{T_2}$$ where $$| \phi \rangle$$ is some pure entangled state.

If I wanted to test some state $$\rho$$, I would have to perform $$\mathrm{Tr}(W \rho)$$. I assume this is done by measuring $$\rho$$ multiple times in the eigenbasis of $$W$$ and finding the expected eigenvalue, and that would be the solution to $$\mathrm{Tr}(W \rho)$$.

1. Is this the way it is done?
2. Therefore, specifically in the above case, it is very much possible to physically apply the witness? (Even though there is a mathematical partial transpose present.)

Just as an example of what I mean, let $$W=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right).$$ As it happens, this is just a swap gate, but ignore this for now. You might find the eigenvectors $$|00\rangle,|11\rangle,(|01\rangle\pm|10\rangle)/\sqrt{2}$$ and measure those expectation values directly. Or, you might write $$W=(\mathbb{I}+Z\otimes Z+X\otimes X+Y\otimes Y)/2,$$ and them you might go off and measure the 3 separate observables $$ZZ$$, $$XX$$ and $$YY$$, those being particularly natural, accessible, things.
Note that there's no problem using $$W$$ to define a measurement. The partial transpose is irrelevant; it's still a Hermitian matrix. The partial transpose just means it might not be a valid state, but being a valid state is irrelevant as a measurement: if we say "do a Z measurement", the Z matrix certainly has nothing to do with being a state. It's just a Hermitian matrix.