For a little less than half the cell cycle, a significant number of genes are represented twice (just before dividing). Does the cell differentiate between these DNA in any way or are transcription rates limited by the amount of polymerase anyway?
I can't unequivocally say, at least without further literature review, that both sister chromatids are transcribed equally everywhere, especially since the cell does have mechanisms to differentiate between the old and new DNA strands (such as methylation state). It would be conceivable to expect, however, that the same regulatory factors that were present before replication have the same effect on newly synthesized DNA as they did on the old. Here are some electron micrographs showing transcription from both chromatids at some locus.
McKnight SL, Miller, OL Jr. 1979. Post-replicative nonribosomal transcription units in D. melanogaster embryos. Cell 17:551-563
Here you can see RNA (solid line in schematic, b) of increasing length branching from both sister chromatids (dotted line):
A similar image but this time showing the replication fork (arrow):
This image shows RNA free gaps (arrows) in a single transcription unit, which are symmetrical between the two chromatids:
From which they conclude:
The observation that fiber-free gaps often occur at symmetrical positions on sister chromatid transcription units suggests that the two identical transcription units respond similarly to the mechanisms that control RNA polymerase chain initiation. These cannot now be specified, but could include the local supply of RNA polymerase or other controlling molecules.