The Mishna in Masekhet Chulin (103b) notes the provision enacted by Chazal forbidding people from eating meat and cheese on the same table. Even if one eats only meat and the other eats only cheese, they nevertheless should not use the same table. This is permitted only if they do not know each other, or if they make some separation between them. The reason for this provision is the concern that friends eating together may share their food with one another, and thus if one eats meat and the other cheese, they may end up violating the prohibition of basar be-chalav (eating meat with milk). The Ran (cited by the Ma’adanei Yom Tov commentary to the Rosh) explains that since meat and cheese are each independently permissible, people are not accustomed to refraining from them. Therefore, a person eating cheese who is offered meat by his friend at the table is prone to inadvertently eating them together.
The Rashash finds it significant that the Mishna speaks in this context specifically of eating cheese, as opposed to drinking milk. The Mishna forbids two people from eating meat and cheese at the same table, but not of a situation where one person eats meat and the other drinks milk at the same table. The reason for this distinction, the Rashash suggests, is because milk, as a liquid, is always contained in its own utensil, and thus necessarily remains separate from the food. It is therefore unlikely that the person eating meat will eat it together with his friend’s milk, and for this reason Chazal forbade only two people eating meat and cheese at the same table. This view is accepted as Halakha by the Darkhei Teshuva (Y.D. 85:5).
Rav Yosef Shalom Elyashiv, however (cited in Umka De-parsha, Parashat Mishpatim, 5774), disagreed with this ruling. He noted that in Masekhet Shabbat (14a), in the context of the laws regarding the status of impurity of foods and beverages, the Gemara specifically states that it is very common for people to drink as they eat. Therefore, Rav Elyashiv claimed, there is indeed reason for concern that a person eating meat with a friend who drinks milk may take some milk from his friend. Rav Elyashiv felt that to the contrary, the Mishna addresses the case of two people eating meat and cheese because it is more obvious that this rabbinic enactment applies when one person eats meat and the other drinks milk. One might have thought that when one person eats meat and the other eats cheese, there is no need to forbid them from eating together, as they are not likely to share their food. Certainly, though, this law applies even when one person eats meat and the other drinks milk.