You would have a slow light house. So on the plane of the beam (say on the plane of the earth's equator, asusming a stationary rotaqting earth) from an arbitrary distance away, you would get a pulsing light. With a perfectly collimated light source and zero divergence, you would get one short pulse followed by a longer time of darkness. With real light sources, but obscenely bright, you would get a fade-in, then a fade out, then a period of darkness. Just light with a light house when viewed from sea. There would also be a Doppler shift in the wavelength in this case, since the light source would be moving toward the observer during the "fade-in" and away from the observer during the "fade-out".
If viewing this situation from a very long distance above the plane of rotation (somehow magically seeing the location of the photons, say from an arbitrarily long distance above the north pole of the earth), then I suppose one could see photons forming some sort of spiral outward from the earth. A photon emitted at noon will have traveled roughly 8 billions miles away from earth in a straight line when, at midnight, another photon is emitted in the exact opposite direction of the first one. Another 12 hours later, the first photon has gone 16billion miles, the second has gone 8 billion miles in the opposite direction, and a third is emitted in the exact same direction as the first. You can see how, if this situation was expanded to a constant stream of photons and could somehow be "viewed" from above, and the locations of photons at a given time were plotted, the locations of the photons at a given time would look like a spiral.
I think part of the OPs question comes from a confusion on how a rotating body and light would interplay. Light's speed doesn't change, so it doesn't depend on how fast the earth was going when the light was emitted. It's not like a tennis ball being spun around an orbit on a string: when you let go of the string, the tennis ball's energy and momentum are conserved, so the tennis ball travels on a tangent to its rotation with the speed it had when you let it go. With a photon, the speed of the photon is independent of the motion of the object emitting the photon. The direction of the photon could depend slightly I suppose (hmm, that's another problem to think about for a minute), but the speed is 100% independent of the lasers motion when the photon is released.
