Learn Icelandic: Við sáumst á hittingnum. - We saw each other at the get-together.

It comes from the active verb sjá (to see). In the past tense, 'we saw' is við sáum. To make it middle voice, you just add the -st ending directly to that conjugated verb, giving us sáumst.

What exactly does the -st ending do to the meaning here?

In this sentence, the -st ending gives the verb a reciprocal meaning ('each other'). While við sáum means 'we saw' (pointing at something else), við sáumst means 'we saw each other'.

Why is 'at the get-together' translated as á hittingnum instead of using the dictionary form hittingur?

The preposition á (at/on) requires the following noun to be in the dative case when you are talking about a stationary location. Hittingur is a masculine noun, and its dative form with the definite article ('the') attached becomes hittingnum.

Is the noun hittingur related to the verb hitta (to meet)?

Yes! Hittingur is a very common, slightly informal noun derived directly from hitta. It is the go-to word in spoken Icelandic for a casual get-together, meetup, or gathering.