First, to contextualize your question a bit. Co-educational college, as we think of them today, didn't fully emerge until the 1950s or 1960s. College as a place where people of all genders and races learned from professors of all genders and races, routinely crossing paths and mingling as a part of daily activities, was fairly uncommon in the era you're asking about.

While there have been co-educational colleges since the mid-1800s, even the most progressive among them maintained the notion of "separate spheres" well into the 1900s. In effect, the thinking went, life after education went and looked very different for men and women so there was no reason for them to learn together - they were headed towards different outcomes. So, men taught men courses, women taught women courses. This type of thinking, a form of benevolent sexism, carried into the conversation about voting rights. It wasn't that women couldn't vote/learn with men, some argued, it was that they didn't need to. But also? Hard thinking was more taxing on the female body and mind, dontcha know. (Written with all of the sarcasm and snide I can muster.)

However, women who wanted to learn the things men learned because they planning on setting their own path moving forward regularly challenged the notion of separate spheres. As an example, Anna Julia Cooper, one of several Black women who attended Oberlin College in the 1880s, simply walked into classes for men and sat down. Cooper, though, was a recent widow and had teaching experience in the content she was taking (Latin, Greek, and rhetoric) and as such, had the social capital to challenge norms at a college created explicitly to be mixed gender and mixed race.

This also contributes to one of the challenges of answering your question as it's hard to know exactly which women were professors in that era. In some cases, including the very progressive Oberlin and Antioch, women with their PhDs were often listed as "Instructors" or "Tutors." There were, of course, clear cases of women professors. Dr. Alice Hamilton, named the first woman professor at Harvard Medical School in 1919 never received a promotion, much less tenure, and was repeatedly approved to teach on a one-year basis. Additionally, she was denied access to a number of perks provided to male professors. These denials were based in both the philosophy of separate spheres and benevolent sexism. She may have been a medical doctor but her male colleagues felt sitting among men during football games was simply too much to ask of her womanly constitution.

So, unfortunately, it's difficult to speak to patterns and generalities given how idiosyncratic each college in that era was. I had trouble tracking down enough examples in the historical record to draw conclusions, alas. I did find some examples of full women professors such as Helen Magill White, the first American woman to receive her PhD on American soil. She opened her own college and staffed it as she saw fit, with women and, I believe, men professors. But her college was a woman's college. Susan Cunningham at Swathmore is another interesting example of a full woman professor but she mostly taught and studied in the late 1800s.

All of that said, there are a few general statements we can make. Odds are good if we're talking about a woman professor at a co-educational college in the early 1900s, we're going to be knee-deep in Quakers, or Friends. Gender equity ran deep into the Friends community and daughters were encouraged to push social boundaries and norms put upon them the non-Friends world. This philosophy is what gave rise to the strong presence of Quaker women among abolitionists, suffragists, and prohibitionists. This isn't to say all women who were raised in Friends traditions were anti-slavery or give-women-the-vote activists. But rather, when we look to progressive movements or those who pushing for social change towards justice and equity, we're likely to find Quaker women. Which is to say, women professors at Friends colleges in that era were less likely to be restricted to a narrow lane around decorum and behavior, more likely to receive permanent positions and live where it made sense for them. In contrast, women professors at non-Friends' colleges and universities in that era were more likely to be relegated to the women's sphere, less likely to be included in professional and cultural norms male professors experienced, and restricted in terms of movement and living.

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Regarding sources, this text, A history of the position of dean of women in a selected group of co-educational colleges and universities in the United States by Lulu Holmes, is adjacent to your question but includes a bunch of examples of how the position Dean of Women [students] evolved as co-educational campuses evolved. Several of the early Deans also taught courses so they weren't full-time professors, per se, but many did appear to have standings on par with male Deans.