Frame challenge: streams' interaction with interstellar matter
You claim that there are streams of particles relatively close to each other, moving opposite directions, each having a speed of no less than 0.01*c with a fixed volume of stream. Skipping the focusing part and acceleration dispersion part (let's say they both consist of atomic hydrogen with speed vectors of exactly zero difference, sent from somewhere to we want to know where - and using neutrons fails due to them being unstable), the streams still face the interstellar matter. Solar wind, comet traces, mere "floating" protons that come into the volume these streams occupy. No matter the density, eventually there will be a collision of one of the streams' particles with a "stationary" object, be it a proton or bigger thing, this doesn't matter. This process would happen with calculatable probability with each particle that is a stream's part over distance travelled, and these interactions result in either stream getting less focused over distance travelled, so their width or diameter would increase. Worse, even if the interstellar space would be devoid of particles, it's not devoid of photons, and these carry impulse as well, AND those impulse vectors are pretty random, as stars are everywhere and the universe background radiation is also omnipresent and will interact with either stream.
So, since the initial trajectories are declared antiparallel, and the widths of both streams continually increase over distance, there will be place where those streams would intersect - provided the streams themselves are long enough. Yet, a speed difference of a mere 1 m/s in a stream, omnidirected, at a base speed of 0.5c (the lower the worse) would cause the stream to grow in diameter over the solar system's diameter by
12*3600*c*1/(0.5*c) = 86400 meters, way more than your 400 meters asserted diameter, and the energy required for a H1 atom to gain 1 m/s sideways velocity is a mere 0.8e-27 J (1.6e-27*1*1/2), or one quantum of frequency about 1.2 MHz. The visible light carries orders more energy per quantum, thus it's only a matter of time spent with the stream until its part would reach the opposite one.
Now, the opposite streams collide, and what would happen? This largely depends on the amount of matter transferred by either, down to almost nothing should the streams' density at collision area lower to at least the density of a nebula (10^6/m^3). Of course, some collisions at double speed would still happen, providing both "stationary" matter and extra photons of high energy in the local area to influence the streams, with them both gaining width the process would cause local density increase together with local temperature increase, but unconfined, it won't cause much harm unless the streams' density would cross some threshold that depends on their contents, speed and initial trajectory distance. The best case would be two largely non-interfering streams of gas flowing opposite directions, with occasional bursts of energy when they would get crossed by something solid, like a comet kernel in an Oort cloud.
The worst case is that they would create an area of superheated gas that would become increasingly opaque to both streams, yet not dispersing fast enough so that the amount of matter in the cross-section of both streams would increase over time. This in turn would make the emerging gas cloud to enlarge with speed about equal to initial of the streams' particles, yet a great part of their kinetic energy would be transformed into local gas's internal energy and outgoing photons, creating a visible effect of an emerging star without actual star being formed. Still anything entering the "walkway" from either side would end up annihilated in that gas cloud, defeating the purpose of them being a media assistant route for interstellar travel. So there would be no reason of those that created the accelerators to ever start creating the streams, since all they would achieve would be an acceleration of star formation somewhere in the distance.