Yes, but not in a really effective manner. Two things occur for an eye to switch to "night mode". The most influential change is that of seeing mainly with the "rod" visual receptors, as you noted. The second, relatively minor change is that of pupil diameter. In the dark, the iris constricts to produce a larger pupil, allowing more light into the eye. This can be done very rapidly, which is why any interruption in this mechanism is negligible.
The important piece of information you need to know is that for most people, each eye's iris acts sympathetically to that of the other. This means that if I were to shine a pen light into your right eye, causing your pupil to constrict, your left eye will do the same, even if no light is entering that eye.
So, if you were to wear an eyepatch over your left eye in a brightly-lit area, and then enter a dark room and remove the eyepatch, you would have far better vision in your left eye because that eye is operating primarily with rods. To answer your question, there was a tiny setback caused by the right eye in terms of pupil diameter, but this would be corrected within a fraction of a second.
Regarding your inference as to a "saturation effect", the answer is yes and no. If you were to wear contacts, with the lens covering the left eye being tinted dark and the lens on the right being clear, then at first it would appear that the light entering the right eye is affecting the vision of the left. This is simply a matter of focus, though. You're distracted by the larger amount of information that needs to be processed coming from the right eye. With proper focus, however, you would see that the left eye is operating independently in terms of light adaptation (aside from pupil diameter, as mentioned).
I hope this answers your question
I've been asked to add references. I'll not provide a reference for the use of rod receptors in low light as you have already acknowledged this fact, but I will provide a reference for the simultaneous pupil dilation. For the saturation effect, you can easily prove this to yourself at home. Skip the contacts and pop one lens out of a dark pair of sunglasses.