Learn English – ‘you by our sides’ or ‘you by our side’

Grammatically, you could use either by considering side as a single side: at the side of the couple, or at the side of each of them individually: at both our sides.

To the couple, or to either member, it is at the side of them together that they would most likely conceptualize in this context: Probably our side because you are likely giving voice to the sentiment of the couple as a pair, which is to say a singularity. And a single entity would have just one side for someone to be at in this context.

This can be illustrated by simply using a word like group.

The group is meeting at 7pm.

Although the group comprises a plural number of members, we think of it as a singularity in this context.

Yet we can also say

The group are dispersing and heading off home in every direction.

In this context, we can't so logically think of a single entity travelling in every direction. That idea emphasizes the group as numerous individuals, so we more felicitously treat it as plural. (Yet it's common, and not necessarily wrong to treat it as singular in this case, too, in American English.)

In British English compared to American English, it is much more common to treat collective nouns as plural.

See the discussion on nouns of multitude under subject-verb agreement here:

https://en.m.wikipedia.org/wiki/American_and_British_English_grammatical_differences