$\newcommand\ket[1]{{|{#1}\rangle}}$The idea for position is the same as for spin. For simplicity, say your particle is moving in 1D and your detectors fire at some fixed time. Say the left detector covers some space region $[l_1,l_2]$ and the right detector covers $[r_1,r_2],$ and these are disjoint. The state (of particle + detectors) right before the measurement may be written $$\ket\psi\ket{\text{ready}_L}\ket{\text{ready}_R},$$ where the first factor is the state of the particle, given by a wavefunction $\psi(x)$ ($\ket\psi=\int_{-\infty}^\infty\psi(x)\ket x\,dx$) and the latter two factors describe the states of the detectors. The state right after measurement is $$\ket{\psi_L}\ket{\text{yes}_L}\ket{\text{no}_R}+\ket{\psi_R}\ket{\text{no}_L}\ket{\text{yes}_R}+\ket{\psi_O}\ket{\text{no}_L}\ket{\text{no}_R},$$
where the states $\ket{\psi_L},\ket{\psi_R},\ket{\psi_O}$ are the "collapsed" wavefunctions resulting from restricting $\psi$ to the appropriate regions. $$\psi_L(x)=\begin{cases}\psi(x)&\text{if $x\in [l_1,l_2]$}\\0&\text{else}\end{cases},\quad\psi_R(x)=\begin{cases}\psi(x)&\text{if $x\in [r_1,r_2]$}\\0&\text{else}\end{cases},\quad\psi_O(x)=\begin{cases}\psi(x)&\text{if $x\notin [l_1,l_2]\cup[r_1,r_2]$}\\0&\text{else}\end{cases}$$ (where again $\ket{\psi_i}=\int_{-\infty}^\infty\psi_i(x)\ket x\,dx$). Note that the overall wavefunction of particle + detectors hasn't collapsed. But the un-collapsed total wavefunction does (and must) encode all the ways the particle wavefunction could have collapsed.
I hope you can see how this relates to the formalism for a spin detector. You write the initial state of the particle as a superposition of (simultaneous) eigenstates of your detector(s)'s observable(s). For each possible outcome on the detector(s), you get a term ("branch") in the wavefunction where the detector(s) shows that result, and it's tensored to the part of the original particle wavefunction that is "consistent" with that result.
Putting the previous paragraph in math terms, you should consider one detector measuring some observable $O$ on a Hilbert space $\mathcal H,$ with possible outcomes (eigenvalues) $\lambda_1,\ldots,\lambda_n.$ (The two detectors in your position experiment can be considered as one detector measuring an observable with eigenvalues $\lambda_1=0$ for "not in left or right region", $\lambda_2=1$ for "in left region", $\lambda_3=2$ for "in right region". In excruciating detail, $O=\int_{l_1}^{l_2}\ket x\langle x|\,dx+\int_{r_1}^{r_2}2\ket x\langle x|\,dx.$) The detector should also have states $\ket{\lambda_1},\ldots,\ket{\lambda_n}$ for when it detects the given result, along with the special initial state $\ket{\text{ready}}.$ When the detector is activated, it sends any state $\ket\psi\otimes\ket{\text{ready}}$ (where $\ket\psi\in\mathcal H$) to the state $$\sum_{i=1}^n[P_i\ket\psi\otimes\ket{\lambda_i}],$$ where $P_i$ is the projection operator onto the eigenspace of $O$ associated to eigenvalue $\lambda_i.$ (In our example, $P_1=\int_{\mathbb R\backslash([l_1,l_2]\cup[r_1,r_2])}\ket x\langle x|\,dx,P_2=\int_{l_1}^{l_2}\ket x\langle x|\,dx,P_3=\int_{r_1}^{r_2}\ket x\langle x|\,dx,$ and these operators "implement" the wavefunction collapse: $\ket{\psi_L}=P_2\ket\psi,\ket{\psi_R}=P_3\ket\psi,\ket{\psi_O}=P_1\ket\psi.$)
