Two observers initially at rest in the expanding Universe

Space and matter

In the standard picture, the reason that space doesn't expand on small scales is not that "it does expand, but gravitational or electromagnetic forces keeps pulling matter together again with a stronger force than the expansion can pull them apart". Rather, space and matter is tied together through gravity:

An initial and still uncertain mechanism called inflation made space expand. The matter (or rather energy at that time) followed along, but its mutual gravitational attraction counteracts the expansion. In some places, there is enough matter that expansion is counteracted completely, keeping matter in these regions from receding from each other. The relevant physical scale for this is galaxy groups.

On the other hand, in some regions there is so little matter that expansion is faster than the average. These regions are the huge voids which are virtually free of galaxies.

Resisting the Hubble flow

Anyway, back to your question: If $A$ and $B$ are $d$ Mpc apart, then the galaxies in the vicinity of $A$ will recede from the galaxies in the vicinity of $B$ with velocity $v = H_0 d$, where $H_0$ is the Hubble constant. How can we ensure that $A$ and $B$ start out with no initial velocity wrt. each other? We can either 1) put $A$ on a spaceship that flies in the direction toward $B$ with velocity $v_mathrm{B} = v$, as measured by a "normal" observer $A'$ which is at rest wrt. some galaxy close to $A$, or 2) we can do the same with $B$, or 3) we can make both $A$ and $B$ fly, e.g. both at $v_mathrm{A} = v_mathrm{B} = v/2$ (where $v_mathrm{A}$ and $v_mathrm{B}$ are directed toward each other).

This puts an upper limit on how far $A$ and $B$ can be from each other: Since it's impossible to travel through space faster than $c$, the maximum distance is (just under) $d_mathrm{max} = 2c/H_0 simeq 8.8,mathrm{Gpc}^dagger$.

Kinematics in an expanding coordinate system

So, what happens to a spacehip traveling through expanding space at a high velocity? We can investigate this using comoving coordinates, i.e. the coordinate system that expands along with the Universe. In this coordinate system, all galaxies lie approximately still (save for their small "peculiar" velocities; of the order a few 100 km s^–1). $A$ and $B$, on the other hand, while their velocity wrt. each other vanish in physical coordinates (by construction of your experiment), they have a non-zero velocity in comoving coordinates. Nevertheless, unless they spend energy to accelerate their spaceships — and this can still only bring them up to a max velocity of (almost) $c$ — their velocity will decrease in comoving coordinates. If in the beginning the fly at (almost) $c$, and if for simplicity we assume that galaxies lie evenly distributed with one galaxy for every 1 Mpc, they will initially pass one galaxy every three million years. However, it will take more and more time to get from one galaxy to the next, since space expands.

In other words, a particle with a velocity through an expanding space will slowly lose kinetic energy, analogous to the redshifting of light.

So, the answer to your question is: If $A$ and $B$ start out sufficiently close to each other, they will eventually meet. But if they start out too far, or if they fly too slowly, they will decelerate and asymtotically go toward rest, ending up being dragged along with the Hubble flow.

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$^dagger$_{Ignoring minor complications like the fast heating of a spaceship traveling through the interstellar medium at a velocity (almost) the speed of light.}

There is some merit to the idea of spatial expansion: FLRW spacetime is conformally flat, leading to a natural notion of freely falling 'rest frames' given by the Hubble flow. The proper distance at constant cosmological time between any points 'at rest' increases, so the space between them is said to expand. This is especially instructive in case of finite universes where this is accompanied by an increase in total spatial volume.

However, what is problematic is that you have to be careful not to forget that despite conformal flatness and a preferred spatial slicing, we're still dealing with a general-relativistic model with nonzero curvature instead of Minkowski space.

For example, your idea about establishing zero proper motion between observers by looking at redshift does not work: Zero redshift means zero relative velocity as evaluated by parallel transport along the light path, which is different from zero change in proper distance at constant cosmological time$^dagger$.

Now regarding your setup, in an expanding universe particles that start out with zero proper velocity are moving towards each other if you take the comoving perspective. Whether this means that they will meet or be pulled apart beforehand (in terms of proper distance) cannot be answered in general as this depends on the initial distance as well as the time evolution of the scale factor. For example, figure 3.1 of the master's thesis linked by Tziolkovski shows one case where the particles never meet, and three cases where they do. In all of the cases, the proper distance ends up increasing, but in the last three cases only after the particles have moved past each other.

As far as a formal solution is concerned, let's see how far we can get without too much effort.

First, the peculiar velocities decrease according to $$ |v_text{pec}| = frac 1 {sqrt{1 + frac {a^2}{{pi_0}^2}}} $$ where $pi_0 = text{const}$.

Given a particle at initial distance $d_0$ from the origin and initial proper velocity $v_0 = 0$, this yields proper velocities $$ v = Hd - frac 1 {sqrt{ 1 + frac {a^2}{{a_0}^2} left( frac 1 {(H_0,d_0)^2} - 1 right) }} $$

Now, let's look at de Sitter spacetime specifically with $a(t) = e^{Ht}$ and assume small initial recession velocities $H_0,d_0 ll 1$.

Taylor expansion yields $$ v(t) approx Hleft( d(t) - d_0,e^{-Ht} right) $$ which is solved by $$ d(t) = d_0,cosh{Ht} $$

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$^dagger$ However, what you can do is split proper velocities into recession and peculiar velocities, which corresponds to a factoring of the frequency shift into cosmological and peculiar Doppler shift. In contrast, if you use the generic approach of parallel transport along the light path, the frequency shift will be wholly Doppler in nature.