For a large proportion of women, dating simply ... stopped. Their boyfriends or fiancés were overseas - instead of being taken out to movies or dinner, they wrote letters and waited for possible furloughs. Single college students usually had to give it up as well, since their campuses had almost no men on them: most co-educational institutions were 75-90% women during the war.

Other women participated in dating much more freely than they would have been able to in the 1930s. (As I've discussed before in this answer, dating outside the home only became an ordinary courtship practice in the early twentieth century.) Between the general upheaval of wartime and the increased independence of many middle-class young women who were now working outside the home to earn a wage, social standards were becoming more permissive. The concept of "going steady" was strong in the 1930s, despite some adult warnings against selfishness and in favor of getting experience with a number of different girls/boys; for young women who were working in areas with an influx of unknown young men (soldiers on leave or at a local base, for instance, or workers who'd migrated), there was greater freedom to ignore the pressure from others to go steady. With the independence of a job, whether in an office or a munitions factory, they could also protest parental restrictions and feel allowed to make their own decisions about sex, or had more opportunities, if they were living on their own. It was this same class of independent working women who had driven the creation of the dating trend earlier in the century (see earlier answer again), so it's unsurprising that they would continue it in the 1940s.

However, don't think that it's a simple narrative of increasing "freedom" and laxity - one standard that shifted in what we could think of as the other way is how partnering in dances worked. In the 1930s, a young man and young woman were supposed to dance together until another man cut in; the only other way they could part is if she saw a group of her friends on the sidelines that she could join. In theory, this meant that part of the young man's responsibility at the dance was to get his friends to show the girl he was escorting a good time, which let her meet more people and feel more socially successful, and also let her escort feel more important due to the apparent competition; in practice, this meant that unpopular girls would be even more neglected as nobody wanted to "get stuck" on the dance floor. Over the course of World War II, due to the lack of American men in most towns, this did a complete 180. It became acceptable and then required for women to compete instead, holding onto the secure date she'd brought to the dance instead of allowing him the liberty of the floor. There simply weren't enough men to allow for the system.