Answer: The Rema (Darchei Moshe YD 92:9) writes that there is a machlokes as to whether one can cook milky food in a pot next to a meaty one. According to the Mordechai (Chulin 691) no flavour gets transferred even if the pots are touching each other while R’ Yisroel Isserlin (Hagaos Shaarei Dura 51:3) maintains that some flavour will be transferred from one to the other. Thus, the Rema (YD 92:8) writes that while pots that touch do not transfer taste, lechatchila one should avoid doing so.
Following this, the Chochmas Adam (74:4) writes that one must kasher a non-kosher tripod that one stands pots on, though bedieved if one didn’t do so, the food and pots are kosher.
R’ Moshe Feinstein (Igros Moshe OC 1:124; YD 1:40; 59; 3:10) however, disagrees, writing that even lechatchila one can use a stove for cooking both milky and meaty food. He explains that the Rema wrote that one should be careful about cooking two dishes adjacent to each other as it is easy for food to boil over. Any food that falls onto the grates themselves would quickly be burned and wouldn’t later contaminate anything else (See Mishna Berura 451:34). One would need to kasher their stove before Pesach, however.
Similarly, the Ksav Sofer (YD 54) writes that one may place both milky and meaty pots on the stove simultaneously even though they may spill onto the stove. He concludes, however, that it is ideal to have separate grates for milky and meaty pots.
In conclusion, if one doesn’t have two stoves, they may cook meat and milk on the same stove, even simultaneously, though one must be vigilant that the pots are kept apart and covered so that nothing can splatter into the other.