If you check a little further south along the border at 57.909489, 27.721441 you'll see a nearly 1km stretch of the Estonian road no 178 passing through the territory of russian federation. Despite being at the EU-russia and NATO-russia border, it is actually allowed to drive there without a visa to russia or any border formalities (there are no border checkpoints). Even riding a bike is permitted, but no stopping or walking. You must cross and re-enter Estonia. This strange border area (called Saatse boot) splits the local municipality in half (the SSR border wasn’t important during soviet times) and otherwise there would be a very long detour on unpaved roads, if you wanted to reach the other side, so for now there is an agreement that this road can be used for Estonian transit.

And it is guarded, even though it looks like nothing is in there and the area is not reachable on the russian road network. Don't let it fool you, they will have field patrols and you will be arrested by the border guard of either russia or Estonia if you commit any border violation. If you're captured by the russians, you will face a trip to the police station in Pskov where you will likely end up being detained for at least several days, or worse.

I drove this road last summer and the locals of course drive it all the time. But I suppose this would also be a potential hybrid warfare target since technically, you're of course at the mercy of whatever the russians will decide to do with you when you drive there. Estonian laws are not possible to apply there and Estonians can't come to save you if the russians were to suddenly decide to scatter some sharp objects on the road that would shatter your tyres, leaving you a sitting duck on unfriendly territory, or if your engine coincidentally happens to break down.

Now I don't know about the status of the lake-exclave, however I wouldn't think that it's any different. If someone is stationed in Dubki, it's 100% certain that they are working for the border guard. It is common sense because there is absolutely nothing else to do in such a remote area within a police state, and it would make no sense to leave the area with no control or else it might as well become a de facto extension of Estonia. An imperialist nation wouldn’t allow that.

Also, the lake would freeze up in the winter and the Dubki exclave would hence effectively be connected to the rest of the russian federation on walkable land. And I fail to see any connection that it might have with refugees or dissidents. These are very remote, nearly unpopulated areas with abysmal roads and marshy terrain. Generally nobody visits (making your movements stick out even more, you will also hear approaching vehicles from a distance, we’re not in a city), but the area would still be patrolled and permissions from the border guard required.