to have each person pay his or her own expenses

The phrase "going Dutch" probably originates from Dutch etiquette. In the Netherlands, it is not unusual to pay separately when dating. The Dutch were already internationally known as scrooges, and English rivalry with The Netherlands especially during the period of the Anglo-Dutch Wars gave rise to several phrases including Dutch that promote certain negative stereotypes. Examples include Dutch courage, Dutch uncle and Dutch wife. The particular stereotype associated with this usage is the idea of Dutch people as ungregarious and selfish.

In Spain, "going Dutch" is attributed to Catalans, due to a stereotype that they are greedy. A stereotypical non-Catalan Spaniard would compete to invite the group.
In Italy, the expression pagare alla romana can be translated as: "To pay like people of Rome" or "to pay like they do in Rome". It has the same meaning as "going Dutch".
Curiously, in South American countries exist the Spanish phrase pagar a la americana (literally "To pay American style") which refers to a treat attributed to North Americans or Canadians.

When you go an a date and everyone pays separately.