A good thing about the shorter epistles like Philippians is that you can read the whole letter in one go, since it doesn't take too long - even with the dense language of Paul.
Go on, give it a try. It will take like less than 20 minutes!
Pause while you read the letter ...
Now having read the whole thing rather than just a chunk or chapter, what is it really about?
The theme of unity and being 'like Christ-minded' runs through the whole letter - put your own interests in 2nd place, following Christ's example. There is also a clever inter-weaving of thanksgiving/rejoicing - with Christ as the focus rejoicing becomes the order of the day.
Now see the progression of 'therefores' and/or 'so thens'. Where do they ultimately lead?
Answer is 4:2, to pleading with Euodia and Syntyche to also be 'like Christ-minded' (same Greek words as 2:2, by the way). The unity affects the gospel work (the mission), and is the key issue. Though the letter is written to the city church, it builds to the unity of these two co-workers. Presumably their disagreement was affecting the whole church, and the mission of the church.
We can conclude that these women were prominent and important to the church and the gospel work. In other words they were leaders. Important enough for Paul to carefully write a letter with their relationship clearly in mind.
That puts Euodia and Syntyche effectively on a par with Timothy, Titus and Philemon.
Paul had various co-workers in the gospel work - they were both male and female. I would imagine that the church today would be the same ... if we take the trouble to read the whole deal, of course.