Probably independently. The first Western monastics were the Desert Fathers, who retreated to the Egyptian desert to avoid temptation and emulate Jesus. Originally, they lived individually as anchorites, only gradually developing the communal monasticism we see today. Internal motivations explain their actions perfectly well, and there's no evidence of Buddhist influence. Indeed, if anything, they may have been influenced by earlier Jewish sects who retreated into the desert, such as the (probably Essene) sect at Qumran that left us the Dead Sea Scrolls.

The Desert Fathers tradition is a rather different from the Buddhist tradition which emphasizes begging for daily sustinance and gathering together for vassa. As far as I know, you don't really get Western monks following anything much resembling that until the rise of the Mendicant Orders (Franciscans and Dominicans) in the late Middle Ages.

Also, it's best to remember that Buddhist montacism arises as a particularly South Asian tradition, not an East Asian one. The Shakyamuni would have fit in very well with other contemporaneous Śramaṇas.