In the following I limit my considerations to 4-point diagrams.
After the introduction of renormalized field operator (in renormalized perturbation theory) $$\phi_r= (\sqrt{Z})^{-1} \phi\tag{10.15}$$ in eq. (10.15) P&S states that "in computing the S-matrix elements, we no longer need the factors of $Z$ in Eq. (7.45);" Eq. (7.45) looks like (I regret not to know well to draw bubbles) $$\langle p_1 p_2|S|k_1k_2\rangle = \tag{7.45}$$ $$ (\sqrt{Z})^4(\mbox{sum of all amputated conn. diagrams with $p_1$, $p_2$ incoming, $k_1$, $k_2$ outgoing}).$$
That means, the factor $(\sqrt{Z})^4$ would no longer appear in eq. (7.45). Actually I don't understand this conclusion.
What I do understand is that when in the LSZ-formula (7.42) the renormalized field operators are used the $Z$-factors disappear:
$$\prod_1^2 \int d^4x_i e^{i p_i x_i} \prod_1^2 \int d^4y_i e^{-i k_i y_i}\langle\Omega|T\{\phi_r(x_1)\phi_r(x_2)\phi_r(y_1)\phi_r(y_2)\}|\Omega\rangle$$ $$\sim \prod_1^2\frac{i}{p_i^2-m^2} \prod_1^2\frac{i}{k_i^2-m^2} \langle p_1 p_2|S|k_1k_2\rangle $$
where for simplicity I neglected the $+i\epsilon$ terms and the limits $p^0_i\rightarrow +E_{p_i}$ and $k^0_i\rightarrow +E_{k_i}$ on the right side of the equation. In the following P&S states that
$$\prod_1^2 \int d^4x_i e^{i p_i x_i} \prod_1^2 \int d^4y_i e^{-i k_i y_i}\langle\Omega|T\{\phi(x_1)\phi(x_2)\phi(y_1)\phi(y_2)\}|\Omega\rangle $$
corresponds to Figure 7.4, i.e. a most general 4-point diagram whose 4 legs contain self-energy bubbles (represented by dark-shaded circles) and the sum of all connected amputated 4-point diagrams in the center. And each self-energy bubble would correspond to a factor like
$$ \frac{i Z}{p^2-m^2}$$
Naively I thought first that this way the $Z$-factors would come in again, but later I thought that in renormalized perturbation theory the self-energy bubbles come also contain the counter terms so that they better correspond to
$$ \frac{i}{p^2-m^2}\tag{10.19}$$
but finally I realized that in renormalized perturbation theory the expression
$$\prod_1^2 \int d^4x_i e^{i p_i x_i} \prod_1^2 \int d^4y_i e^{-i k_i y_i}\langle\Omega|T\{\phi_r(x_1)\phi_r(x_2)\phi_r(y_1)\phi_r(y_2)\}|\Omega\rangle$$
should be considered instead. Looking at this expression I am no longer sure whether it would be still correspond to figure (7.4), i.e.
$$\frac{iZ}{p_1^2-m^2}\frac{iZ}{p_2^2-m^2}\frac{iZ}{k_1^2-m^2}\frac{iZ}{k_2^2-m^2}\cdot (\mbox{sum of all amputated conn. 4-point diagrams})$$
which only seems to hold for non-normalized field operators. For renormalized field operators the $Z$'s in the precedent expression certainly disappear, but does the $$(\mbox{sum of all amputated conn. 4-point diagram})$$ the same? It is not clear for me. This is, however, the prequisite for the validity of eq. (7.45) without $Z$'s, the equation I cited at the beginning of my question.
I would be grateful if somebody with more insight could explain it to me.
