The stalk of the structure sheaf on the étale site at a geometric point  is the strict henselization of the corresponding local ring, cf. [Section 4 of Milne's lecture notes on étale cohomology][1], or EGA IV. 

There are two things to say regarding the  commutation issue raised in anon's comment (which you can view as a continuity condition if you want): 

- in the case  of Prop 6.12 of the lecture notes on motivic cohomology, $F=\mathbb{Z}_{\operatorname{tr}}(T)$ for $T$ a scheme of finite type. Because of the finite type assumption, every correspondence $\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}}\to T$ is in fact represented by a correspondence  $U\to T$ where $U$ is an étale neighbourhood of $x$ in $X$; therefore the stalk of $\mathbb{Z}_{\operatorname{tr}}(T)$ at $x$ can be computed as $\mathbb{Z}_{\operatorname{tr}}(T)(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. (See also Exercise 1.13 in the lecture notes.)

- in the argument for the rigidity theorem, the functors that are being considered  (i.e. presheaves with transfers) are defined for schemes of finite type - in some sense, there is actually no definition of $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. So the functors are extended to local rings (or more general, essentially finite type schemes) by the definition $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})=\operatorname{colim}F(U)$ where $U$ runs over all étale neighbourhoods of $x$ in $X$. The necessary continuity problem is defined away... (As a side note: this is not possible  in the original application of the rigidity theorem to K-theory because K-theory is defined for all rings. Fortunately, K-theory commutes with directed colimits as in the definition of the local rings, so the continuity requirement is again met.)

  [1]: http://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf