Between power limitations and aperture-limited diffraction I just can't see how it's possible.
In the video of the Keynote Speaker (and former NASA ARC director) Pete Worden (also here)'s talk Breakthrough Discuss 2016 – Breakthrough Starshot Challenges the challenges of using the deployed photon sail (for the initial laser-based acceleration) as a reflector "dish" for an optical transmission back to Earth is discussed a bit. The speaker felt that this might be one of the most difficult technological challenges in the short term, possibly with no current or near-future solution.
Getting a tiny spacecraft to somehow get the 300 angstrom thin membrane to form an accurate optical surface to collimate an optical transmission does indeed sound pretty difficult.
Let's use a burst of 100W and a 10 meter reflector and a distance of 5 light years (5E+16 meters).
800nm photons from a simple III-V solid state laser have an energy of about 2.5E-19 Joules each, so that's 4E+20 photons per second. An Airy disk half-width of 1.22 $\lambda$/d of 8E-08 radians means the footprint at Earth has a radius of 8E-08 x 5E+16 = 4E+09 meters. The radius of the Extremely Large Telescope's aperture is about 20 meters, so the ratio of the areas is 2.5E-17.
Since we're transmitting 4E+20 photons per second, that's about 10,000 photons per second at the receiver!
Your milage may vary, but in terms of energy, is quantitatively realistic.
However, from an engineering point of view, one would have to back off from this starting point. Forming a 10 meter parabola with $\lambda$/4 from a 300 angstrom film automatically using a chip-size satellite is quite a challenge - probably nobody really knows how it might be done at the moment.
Pointing is also important; what kind of star cameras could one include within a tiny chip that would provide 0.02 arcsec pointing accuracy is currently unknown.
So the real challenge will be building something even remotely close to this. In order to transmit meaningful data, millions of photons would have to be collected. Probably dedicated, lower-tech photon collectors in the microgravity of space would be better than expensive Earth-based telescopes.
The inaugural 2016 Breakthrough Discuss Workshop was held April 15-16, 2016 at Stanford University and sponsored by the Breakthrough Initiatives and the Stanford Physics Department. Breakthrough Discuss is a forum for scientists to present and discuss leading edge ideas in space exploration. The 2016 workshop focused on the following three areas:
Optical SETI and the Detectability of Directed Energy Systems – Chaired by Jill Tarter
Exo-planet Detection Programs Focused on Alpha Centauri – Chaired by Olivier Guyon
The Sun as a Gravitational Lens – Chaired by Avi Loeb
Either we would need to launch waves of spacecraft to form a web of relay signals, or we would have to manually retrieve spacecraft sometime in the future. Power could easily be solved by using the same principle of 'torch ships' (that is, breaking down the materials of the ship into energy). However, any communication would be next to impossible.
I think that the problem of communication could be done without needing an interstellar power source. What if the super powered telescopes of earth are used to retrieve signal from the ship, and the ships signals are how it reflects incoming beams from a high-powered emitter on earth. Or Since they are micro-ships, have a specific maneuver or pack pattern as a signal.
Power limitations are tricky, in particular choosing a power source whose mass isn't much more than that of the spacecraft. Nonrechargeable batteries have too short a shelf life. Internal combustion engines that small have been tried, and abandoned. A milligram-scale RTG + supercap, maybe with a Diamond battery, is the only way I know to sustain 40 Kbps to the terrestrial 30 m reflector mirror.
Details and references are here: https://space.stackexchange.com/a/17072/1235