We consider the identity testing problem in the context of high-dimensional distributions under access to conditional samples. Given as input a visible probability distribution and access to a sampling oracle for a hidden distribution, the goal in identity testing is to distinguish whether the visible and hidden distributions are the same or not. When there is only access to full samples from the hidden distribution, it is known that exponentially many samples (in the dimension) may be needed for identity testing. To cope with the computational intractability of identity testing, previous works have studied identity testing with additional access to various "conditional" sampling oracles. We consider a significantly weaker conditional sampling oracle and provide a complete computational and statistical characterization of the identity testing problem in this model.