First, the non-locality in Bohmian mechanics is not due to entanglement or to measurement. It is simply baked into the Bohmian approach from the beginning. It is true that is a hidden variable theory that gets around Bell's inequalities by being a non-local hidden variable theory. But, the non-locality is not "due to" entanglement.

Bohmian mechanics for $N$ particles in $3$ spatial dimensions is based on $3N+1$ equations in $3N+1$ variables.

• $3N$ of the variables are the trajectories of the $N$ particles, $\vec{q}^k(t)$, where $k=1, 2, \cdots, N$, and the vector symbol $\vec{}$ indicates that $\vec{q}^k$ is a vector in ordinary 3-space.

• The remaining variable is the pilot wave $\Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t)$.

First, there is the equation for the pilot wave $\Psi$, which is just the ordinary Schrodinger equation $$ i \hbar \frac{\partial \Psi(q, t)}{\partial t} = H \Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t) $$ Then there are equations of motion for each of the $N$ particles $$ m_k \frac{d \vec{q}^k}{dt} = \hbar \nabla_k {\rm Im}\ {\rm ln} \Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t) $$ Finally, the Born rule is recovered by arguing that the distribution of particles reaches an equilibrium distribution $|\Psi|^2$.

The nonlocality in Bohmian mechanics arises from the second equation; the fact that the equation of motion for (say) $\vec{q}^1$ at time $t$ depends on the positions of all the other particles $\vec{q}^2, \cdots \vec{q}^N$ at time $t$. Whether you want to say $\Psi$ is a "medium through which the non-locality travels" is up to you... in light of the lessons of special relativity and the aether, I am inclined not to introduce a loaded term like "medium" and to simply trust the equations to describe the non-intuitive structure they are describing.

First, the non-locality in Bohmian mechanics is not due to entanglement or to measurement. It is simply baked into the Bohmian approach from the beginning. It is true that is a hidden variable theory that gets around Bell's inequalities by being a non-local hidden variable theory. But, the non-locality is not "due to" entanglement.

Bohmian mechanics for $N$ particles in $3$ spatial dimensions is based on $3N+1$ equations in $3N+1$ variables.

• $3N$ of the variables are the trajectories of the $N$ particles, $\vec{q}^k(t)$, where $k=1, 2, \cdots, N$, and the vector symbol $\vec{}$ indicates that $\vec{q}^k$ is a vector in ordinary 3-space.

• The remaining variable is the pilot wave $\Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t)$.

First, there is the equation for the pilot wave $\Psi$, which is just the ordinary Schrodinger equation $$ i \hbar \frac{\partial \Psi(q, t)}{\partial t} = H \Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t) $$ Then there are equations of motion for each of the $N$ particles $$ m_k \frac{d \vec{q}^k}{dt} = \hbar \nabla_k {\rm Im}\ {\rm ln} \Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t) $$ Finally, the Born rule is recovered by arguing that the distribution of particles reaches an equilibrium distribution $|\Psi|^2$.

The nonlocality in Bohmian mechanics arises from the second equation; the fact that the equation of motion for (say) $\vec{q}^1$ at time $t$ depends on the positions of all the other particles $\vec{q}^2, \cdots \vec{q}^N$ at time $t$. Whether you want to say $\Psi$ is a "medium through which the non-locality travels" is up to you... in light of the lessons of special relativity and the aether, I am inclined not to introduce a loaded term like "medium" and to simply trust the equations to describe the non-intuitive structure they are describing.

In response to some discussion in the comments, I'll note my discussion is essentially just a summary of the wikipedia article.