It is technically ambiguous, but I think most people, by far, would take the former interpretation (you are not distancing from them; you, as a group, are collectively distancing from others).
I say it's technically ambiguous, because the only thing that is clear is that you are doing this activity (social distancing) with them. How you do this activity, and the role you play in relation to each other, is the part you have left open to interpretation with this sentence structure, and the term social distancing isn't quite well-established enough in the language to have built its own special relationship with the word with as so many other verbs have.
To understand this better, let's try a different sentence with a similar structure: I am playing chess with Jessica. Technically, this could mean:
- Jessica and I are playing chess, and Jessica is my opponent
- Jessica and I are playing chess with a third person (or computer, etc.), and we are working together to defeat them, even though it is traditionally a two-player game
- Jessica and I have decided to spend some time in each other's company, each playing chess against computers or online opponents, etc., as a shared activity, while talking and having beer
- Jessica and I are on a team going to some sort of team chess competition
Because chess is understood to be a two-player, competitive game, the former interpretation would be the default understanding, absent some sort of context that complicates it. If you were describing one of the other scenarios, you would generally make it explicit to avoid the confusion.
If you are social distancing, this is generally something you do in relation to everyone, not specific people. The reason the term social distancing exists is to make it clear that it is people that you are keeping your distance from, and if you are talking about specific people then the word social isn't really necessary. I might say my brother is in quarantine or that I am keeping my distance from my brother, but it sounds a bit odd to say that I am social distancing from my brother.
This, then, provides a context from which we can derive a default expecation of what with would refer to here. If social distancing is an activity that isn't in relation to specific people, then when I am social distancing with my brother, the default implication would be that it is an activity we are doing together in relation to all other people, rather than in relation to each other. And in today's context, the most natural way to understand that would be that we are actually making an exception and not distancing from each other, and thereby forming a household/bubble. So I would take the phrase I am social distancing with my brother to be a shorthand for I am social distancing, with the exception of my brother, as we have formed a bubble/household that acts as a mutual pact wherein we will do what we can to minimise our likelihood of being infected, with the knowledge that we would likely infect each other.
This ambiguity is very much built into the word with, and you can see that if you compare what you would think of as the default interpretation for each of these sentences:
- I spent most of Saturday arguing with Jeremiah
- I have stolen bread with Julia
- I have broken bread with Jedidiah
- I do not enjoy fencing with Junko
- I am WhatsApping with Josephine as we speak
- I served on the HMS Victory with Joshua
- I fought with Jericho several times
- I ran along the Thames with Jacqueline
- I was in a race with Jamilah to finish the project
- I circumnavigated the globe with Jacopo
- It was when I quarantined with Jasper that our relationship really took hold
In each of these, there's a sort of default expectation of what the with implies in relation to the activity. You can sort of visualise them, and in some cases there may be an antagonistic relationship, a friendly competitive relationship, and in others a cooperative one. In each case, technically other interpretations are possible, but they would all be edge cases and would require some context or disambiguation for them to become reasonable interpretations.