Being relatively geographically close, I'm wondering if there was perhaps some sort of mutual influence between the two literary traditions. Is there any evidence of something like this? If not, could some of the legends have sprung from a common source?
This question is best answered by the work of the late Bo Almqvist (1931-2013), former director of the Department of Irish Folklore, a specialist in Celtic-Scandinavian contact as it manifested in folklore. Of particular interest is his collection of essays, edited by Éilís Ní Dhuibhne and Séamas Ó Catháin, Viking Ale: Studies on Folklore Contacts between the Northern and the Western Worlds (Aberystwyth: Boethius, 1991). In particular is his essay Entitled Scandinavian and Celtic Folklore Contacts (the following from page 10):
Some elements of the [Edda and Saga] tradition were common to all the Scandinavian countries and the Viking colonies in the west, and some also had their roots in a tradition common to all the Germanic-speaking peoples. But heroic tales were also a favourite literary and oral genre among the Gaels, and among them stories about Cu Chulainn, Fionn Mac Cumhaill, and other heroes, can still be heard from the mouths of living storytellers.
Anybody who takes the trouble to compare these literary and oral stories in the Ulster and Fenian Cycles with those found in Scandinavian and Germanic tradition about Sigurthr the Slayer of Fafnir, Helgi Hundingsbani, Ragnarr lothbrok etc. cannot but be struck by the many strong similarities. The pitfalls for those who would explain these similarities are many and varied, however. The pattern of heroic life is similar all over the world; some themes and motifs, such as the father-and-son combat, seem to be a common Indo-European heritage; exchanges of tradition may have taken place between the Celtic and the Germanic peoples on the Continent, later to spread to Wales, Ireland and Scotland on the one hand and to Scandinavia on the other - to mention just some of the possibilities. Nevertheless there are a number of instances where similarities between Gaelic and Scandinavian heroic tales are so close that the likeliest explanation is direct loans in either direction; and this likelihood is sometimes strengthened by the absence of the motifs outside the Celtic-Scandinavian area.
Sorry for the long quote, but Almqvist was the master on this topic, and we do well to sit at his knee and hear him speak. In short, there is some evidence of diffusion and some hint of common older sources. Almqvist was an exacting scientist who did not suffer speculation gladly, so much of his work raises more questions than it answers, but he does serve as a means to begin the process of sorting all of this out.