Why do two vertex need to be located at the same position for momentum shell RG of $\phi^4$ theory? When we evaluate the momentum shell RG for $\phi^4$ theory (assuming in Euclidean space):
$$S[\phi]_{E}=\int d^{D} x\left[\frac{1}{2}(\partial \phi)^{2}+\frac{1}{2} r \phi^{2}+\frac{1}{4 !} g \phi^{4}\right]$$
then we integral out the fast mode $\phi_f$ whose momentum above the cutoff $\Lambda$, which gives the one-loop correction to $g$ as following:

According to the Feynman rules in the position space, this diagram should be expressed as:
$$\left(-\frac{1}{4} g\right)^{2} \int d^{D} x \int d^{D} x^{\prime} G_{0}^{f}\left(x-x^{\prime}\right) G_{0}^{f}\left(x-x^{\prime}\right) \phi_{s}(x) \phi_{s}(x) \phi_{s}\left(x^{\prime}\right) \phi_{s}\left(x^{\prime}\right)$$
and it is different from the initial $\phi^4$ term, i.e. we need to integral both $x$ and $x'$. Thus, I think we need to add a delta function like $\delta(x-x')$ to keep two forms consistence, but I don't know the reason of it. 
Also, when I add $\delta(x-x')$ by hand, the expression of this diagram will be:
$$\left(-\frac{1}{4} g\right)^{2} \int d^{D} x G_{0}^{f}(x-x) G_{0}^{f}(x-x) \phi_{s}(x) \phi_{s}(x) \phi_{s}(x) \phi_{s}(x)$$
after Fourier transformation:
$$\begin{array}{l}=\left(-\frac{1}{4} g\right)^{2} \int d^{D} x \int d^{D} p \int d^{D} q \int d^{D} k_{1} \int d^{D} k_{2} \int d^{D} k_{3} \int d^{D} k_{4} G_{0}^{f}(p) G_{0}^{f}(q) \phi_{s}\left(k_{1}\right) \phi_{s}\left(k_{2}\right) \phi_{s}\left(k_{3}\right) \phi_{s}\left(k_{4}\right) e^{i\left(k_{1}+k_{2}+k_{3}+k_{4}\right) x} \\ =\left(-\frac{1}{4} g\right)^{2} \int d^{D} p \int d^{D} q \int d^{D} k_{1} \int d^{D} k_{2} \int d^{D} k_{3} G_{0}^{f}(p) G_{0}^{f}(q) \phi_{s}\left(k_{1}\right) \phi_{s}\left(k_{2}\right) \phi_{s}\left(k_{3}\right) \phi_{s}\left(-k_{1}-k_{2}-k_{3}\right)\end{array}$$
which is different from the result in the reference 1 and 2. In the other words, is just seems there are some redudent degree of freedom.
Reference


*

*P453, Altland, Condensed Matter Field Theory

*eq. 4.22, Sachdev, Quantum phase transition 

 A: What is produced by integrating $\phi_f$ is indeed a term $A_{\rm true}$ which looks like a $\phi_s^4$ except for the nonlocality due to two fields being at $x$ and the other two at $x'$. Now if you add by hand a delta function forcing $x'=x$ then what you get is another expression $A_{\rm approx}$. In fact, what one has to do simply is to write
$$
A_{\rm true}=A_{\rm approx}+(A_{\rm true}-A_{\rm approx})\ .
$$
The correction to the coupling comes from $A_{\rm approx}$ which is a true local $\phi^4$ vertex. But then what happens to the correction? One can in fact expand it using a Taylor series in $x'-x$, or simply use the fundamental theorem of calculus. In any case this generate terms like $\phi^3\partial\phi$ which contain derivatives and thus increase the scaling dimension. What you get are irrelevant terms in the RG sense. What makes $A_{\rm approx}$ a good approximation is that the "hard" dotted lines due to the fast field decay at a small scale say $L_{\rm small}$ while the slow field does not vary much on that scale.
For more details and less handwaving, see:


*

*Section II.2 of the book "From Perturbative to Constructive Renormalization" by V. Rivasseau,

*Section 14 of the lectures "Introduction to the Renormalization Group" by A. Kupiainen.

