I sort of addressed this in an earlier answer to What did guys do to express interest in women before telephones became widespread in the 1900s?

This is an interesting question because of the underlying assumption - that it was acceptable to strike up an acquaintance on the street out of a desire for further emotional intimacy. This is actually a very modern stance: it was not in the least proper, historically. Dating did not really exist prior to the twentieth century. A woman was typically courted for the purposes of future marriage by a man known to her and her family - when she was only just eligible for courting, her mother would invite suitable partners, and later she could invite people she'd been introduced to; someone who wanted to become more closely acquainted with a young woman would have to know another person in her circle and get them to introduce him. Instead of going out alone with a suitor, he would call on her at home and interact with her in the parlor, potentially while one or more of her family members were there. Opportunities to be alone - walking to or from church, being unsupervised in the parlor, etc. - could come with trust gained from that family. (The poorer and more crammed-in the family, the less the "calling" standard held, simply because there might not be space in the home, though anecdotes in autobiographies tell of family members sent to wait in the kitchen so the parlor could be used by a sister.)

Dating culture only starts to appear around the turn of the century, in two groups of women: workers and collegiates. In the 1880s and 1890s, women started to join the workforce (as factory workers, as saleswomen, as secretaries, as waitresses, etc.) in larger numbers than before, and to do so to support themselves rather than their families - they were, in fact moving out of the family home and into their own rooms in boardinghouses. According to Moira Weigl in Labor of Love, by 1900 half of the population of American women were working outside the home, and were free and able to meet young men they didn't already know. However, since they were paid half (or less) of what men in similar positions were being paid, they also had very little discretionary income - there was no chance to go out and do anything fun unless they were being shown a good time by a young man, and just as this implies, this could be a slightly seedy transaction. A "charity girl" was a young woman who was "treated" (taken out) by a man in exchange for sex, legally safe from being prosecuted for prostitution, but considered essentially the same as a prostitute by the parts of society not involved with the dating world. College students were in a similar but much better situation. Like young working women, they had independence from the family home and were plunged into a world with many young men they could introduce themselves to; because they were at college, though, they were usually from an affluent enough background that they weren't so destitute. Both groups would go out to dance halls, music halls/vaudeville, and movie theaters instead of the family parlor. In the 1910s and 1920s, the concept of "public courtship" began to seep into ordinary late-teenagehood.

But, anyway, your specific question. What contemporary fiction and newspaper articles show is that these couples would typically meet up because they were regularly in the same place at the same time, or at least one of the members could be counted on to be so. The women in one factory and the men at another would come together in the street at closing time. A shopgirl would be on the same counter every day and could be found by a customer who was trying to court her. The woman ironing at the front of the laundry was found by men leaving their shirts to be washed and starched. Whereas women had previously taken the lead in inviting men to pay calls, in the new dating culture men were, if not responsible for starting the relationship, given more power in the initial stages. In arranging to take a young woman out after talking to her at or after work, he could find out her address (in order to pick her up and take her home) and then move on from there.

Further reading:

Beth L. Bailey, From Front Porch to Back Seat: Courtship in Twentieth-Century America (Johns Hopkins University Press, 1989)

Moira Weigl, Labor of Love: the Invention of Dating (Farrar, Straus and Giroux, 2016)