Based on the most recent poll on the NSA counter-terrorism program showing thin Democratic support and broad Republican opposition, it’s obvious that, yes, Republicans generally trust Republican governments to carry out surveillance efforts and Democrats generally trust Democratic governments.
The difference here, however, is that Republicans supported the surveillance operations when they were conducted without warrants or any oversight whatsoever. On the other hand, Democrats support surveillance in a time when judicial oversight exists on several levels.
Yes, Greenwald and Snowden were successful in one of their goals: scrambling the left-right paradigm on this issue. But it’s unclear at this point whether NSA-opponents on both sides will team up to do anything positive. Honestly, I really, really doubt it.