Cold food does not transfer the taste. Only when the taste of meat or milk is infused in the other is this food prohibited. Halakhicly taste is only transferred in four ways: 1, through heat; 2, through fat; 3, through salting (chemical heat); and 4, through pickling (the milk and meat are in a liquid for at least 24 hours straight) Daf TB Khullin 107 discusses do you need to shopping bags, one for meat products and one for dairy products, when returning home from the supermarket.
“MISHNA: A person may bind meat and cheese in one cloth, provided that they do not come into contact with each other….
“GEMARA: The mishna teaches that one may bind meat and cheese together in one cloth, provided that they do not come into contact with each other. The Gemara asks: And if they come into contact with each other, what of it? It is a case of one cold food in contact with another cold food, and they would not absorb substances from one another. Abaye said: Granted that cold foods do not require the peeling of the place where they came into contact, as they do not absorb substances from one another. Nevertheless, don’t they require rinsing in water? The Sages therefore decreed against the contact of even cold meat and cheese, lest one come to eat them without rinsing them first.” (Sefaria.org translation)
Obviously we try to keep meat products and milk products separate as a fence around the Torah prohibition. Nevertheless, we must keep in mind that taste is only transferred in the four above methods. When cold meat and cold cheese accidentally touch each other, no taste is transferred and both are permitted. All one has to do is rinse the meat and cheese under running water to remove any of the other food particles clinging to it. Once you do that, both the milk and meat products may be eaten.
Today some people separate meat and
milk products so completely their homes have two sinks and two dishwashers.
They may even have two refrigerators. In my opinion none of this is required in
order to keep a kosher home. When I was
a rabbinical student I studied the laws of kashrut with Dr. José Faur z”l. He
once said in class that Jews today are more strict in their observance of
kashrut and need be and more lenient in their observance of Shabbat. He was
right then and still right today.