It is prohibited for two people to eat milk and meat at the same table (שו”ע יו”ד פח:ב). However, this prohibition only applies if the two are comfortable enough with each other to eat from one another’s plate without asking or being ashamed. If the two do not know each other there is no issue even though they are eating together, such as in a hotel dining room or at a cafeteria table. One should be careful to always have separate dips for meat and milk, since this can easily lead to mistakenly eating milk and meat together.