There is a solution within JIRA which is to copy the task and then move it to the second project. An alternative option is to use the Clone+ plugin from Bob Swift which will allow you to do these two steps in a single action (and other things too).
But this raises another question which is how will you ensure that both stories are updated in sync? This may be the case for you having a single board for both projects and then using quick filters to swap between issues relevant to one project or the other.
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