Eusebius's Ecclesiastical History 4.22.7-8 tells us that Hegesippus was a Hebrew, and from 4.22 we learn that he travelled to Rome via Corinth:
Hegesippus in the five books of Memoirs which have come down to us has left a most complete record of his own views. In them he states that on a journey to Rome he met a great many bishops, and that he received the same doctrine from all. It is fitting to hear what he says after making some remarks about the epistle of Clement to the Corinthians. His words are as follows: “And the church of Corinth continued in the true faith until Primus was bishop in Corinth. I conversed with them on my way to Rome, and abode with the Corinthians many days, during which we were mutually refreshed in the true doctrine.
This journey was in about A.D. 160 because Eusebius says that it was at the time of Anicetus. Hegesippus passed through Aegean churches such as Corinth. Presumably these churches had preserved Paul's letters and his legacy. Since Hegesippus was in agreement with the doctrine held by all the bishops that he met, we can assume that he was no opponent of Paul's influence. Indeed, his statement that Corinth "continued in the true faith" suggests that he believed that Corinth was already in the true faith, at least after Clement's correcting letter (and Clement was also a fan of Paul). So Hegesippus endorsed Paul's legacy.
But Hegesippus was also a huge admirer of James:
The manner of James’ death has been already indicated by the above-quoted words of Clement, who records that he was thrown from the pinnacle of the temple, and was beaten to death with a club. But Hegesippus, who lived immediately after the apostles, gives the most accurate account in the fifth book of his Memoirs. He writes as follows: “James, the brother of the Lord, succeeded to the government of the Church in conjunction with the apostles. He has been called the Just by all from the time of our Saviour to the present day; for there were many that bore the name of James. He was holy from his mother’s womb; and he drank no wine nor strong drink, nor did he eat flesh. No razor came upon his head; he did not anoint himself with oil, and he did not use the bath. He alone was permitted to enter into the holy place; for he wore not woolen but linen garments. And he was in the habit of entering alone into the temple, and was frequently found upon his knees begging forgiveness for the people, so that his knees became hard like those of a camel, in consequence of his constantly bending them in his worship of God, and asking forgiveness for the people. Because of his exceeding great justice he was called the Just, and Oblias, which signifies in Greek, ‘Bulwark of the people’ and ‘Justice,’ in accordance with what the prophets declare concerning him. (Eusebius, Ecclesiastical History, 2.23.3-7)
So, Hegesippus endorsed James too. It is therefore hard to believe that there was a lasting doctrinal schism between Paul and James, yet this seems to be the view of many. Neither Hegesippus, nor Luke saw any rift between the two men. The rift is created only by those who misread Galatians. See here.
Incidentally, Peter Kirby has an interesting post on Hegesippus. He suggests that some words of Hegesippus have been misattributed to Josephus by Clement of Alexandria and by Origen. I think this kind of confused attribution is very possible, especially if Hegesippus's Hebrew name was Joseph.