Answer: The Gemara (Pesachim 76a) teaches that there is a machlokes as to whether we are concerned about aroma, reiach, being transferred between two foods that are cooked together in an oven. While Tosafos (Pesachim 76b) follows the stricter opinion, Rambam (Maachalos Asuros 15:33) and the Shulchan Aruch (YD 108:1) write that reiach from a non-kosher dish does not render the other non-kosher, though one mustn’t cook such foods together lechatchila. The Rema adds that the same halacha applies to cooking a meat dish alongside a milky dish. Thus, one cannot cook a milky dish and meat dish in the oven simultaneously.
The Mishna (Machshirim 2:2) teaches that hot steam, zeiah, can render food tamei. Following this, the Rosh (Teshuvos 20:26) writes that one can’t cook a meaty dish above a milky one as the steam from the lower dish is considered to be like milk. The Shulchan Aruch (YD 92:8) paskens this way, though the Rema writes that if one of the pots is covered, then the food is kosher, though urges one to be stringent in this regard.
There is a machlokes as to whether one can use an oven consecutively for meat and milk.
R’ Mordechai Yaakov Breisch (Chelkas Yakov YD:23) writes that one cannot use the oven freely for both as the zeiah remains in the walls of the oven. Similarly, R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 5:20) encourages one to have separate ovens (See Yabia Omer YD 5:7). Otherwise, one must double wrap milky food in a meaty oven (or vice-versa) or kasher it in between.
R’ Moshe Feinstein (Igros Moshe YD 1:40; 1:59) advises that one designate the oven meaty (or vice-versa) and ensure that any milky food one places in later is tightly covered to ensure that no zeiah can enter. As solid dishes don’t produce any real zeiah, one would be allowed to bake a milky dish uncovered in a meaty oven. One must wait until the oven cools down and ensure that the oven is clean.
The Aruch Hashulchan (YD 92:55) writes that as our ovens are ventilated there would be no issue of either reiach or zeiah, and one could, therefore, use the oven simultaneously for meat and milk.
In conclusion, while it is preferable to have separate ovens for meat and milk, one can use one oven for both. Ideally, one should designate their oven either meaty or milky, and wait until it cools down before putting the other dish in. Unless it is a dry dish, one should cover it well and ensure that the oven is clean.