I am working on a project in which an XOR gate is being built by having two hammerhead ribozymes simultaneously cleave each other when both are expressed, but have these ribozymes cleave a target transcript when expressed alone. Has it ever been shown in the literature that a hammerhead ribozyme was able to cleave the flanking arms of another ribozyme?
[ For the sake of answering a question that is long unanswered ]
Yes, in principle they can.
This may be realized only when ribosomes have higher affinity for each other than for the target. As you said in the comment, inserting few mismatches in the ribozyme-target interaction may work. You may need to notice that since the two ribozymes are complementary to each other but cleave the same target, the target sequences will form a set of two mutually complementary regions. If separated well enough, there may not be much of a problem. From your design strategy I guess that UTRs are the best place to keep the ribozyme target sequence. If you keep both the target sequences in the 3'UTR (lets say), then they may form a hairpin structure, which may interfere in the targeting. So you can insert mismatches that also destabilize the possible hairpin.
Not yet. Self cleaving examples are there but not mutually cleaving ones.