Why do electrodes need to share an electrolyte in a lemon battery? From what I understand of a lemon battery, there are two electrodes one zinc, and one copper. The electrodes are then both placed separated in the same electrolyte, in this case a lemon. The Zinc dissolves into the electrolyte oxidizes and loses 2 electrons. Those 2 electrons stay on the zinc electrode. On the copper electrode 2 h+ combine with 2 electrons from the copper and then form h2. the h2 bubbles up and leaves the electrolyte. The extra electrons from the zinc then move to the copper when there is a connection between the anode and cathode. This reaction continues until the zinc is too dissolved or if there is not enough h+ in the electrolyte to take electrons from the copper.

If my understanding is correct (please correct me if I'm wrong), then why can't each electrode have is own electrolyte, completely separated. For example the anode in half a lemon and the cathode in the other half. Both half's separated by air.
Like @Pritt said, the circle has to be closed, 2 electrons are traveling from one half-cell to the other, means 2 negative charges are missing, the half-cell needs to replace it by other 2 negative ions or send 2 positive charges, since charges cannot exist separately.