Here is a link to Sidney Coleman's seminal paper Fate of the false vacuum: Semiclassical theoryin the KEK library. He discusses a setting that is not eternal inflation, but where the false vacuum has zero energy density and essentially gives Minkowski space as the solution. In figure 4 you can see exactly the acceleration of the wall mentioned by @kimbyd.

Side note: This paper contains one of my favourite quotes across all physics papers that I have read. It highlights a somewhat morbid sense of humour in Coleman (my clarifications in square brackets):

I found this paper which attempts to do a detailed treatment of the dynamics of domain walls: https://arxiv.org/abs/0811.0866

It's quite dense, unfortunately, but it describes a picture where once a new domain starts to form, it transitions from no acceleration to constant acceleration, which means the domain wall asymptotically approaches the speed of light.

This is off the top of my head, but it probably stems from the boundary conditions. Consider that you're transitioning from a state with ##H^2 = \Lambda## (with the appropriate constant conversion factor I'm not going to bother with right now) to a state with a lower cosmological constant. This results in an immediate decrease in the energy density associated with the cosmological constant, but the rate of expansion would (initially) remain the same. If this energy density didn't go into any other matter/radiation field, or if there was an equal amount of energy density in matter and radiation produced but not all of it ended up within the expanding bubble, then there would be a discrepancy that would show up as negative curvature.

So if I were to guess, the details of the spatial curvature are all about how the transition to a lower vacuum energy interacts with matter/radiation, and how that matter/radiation is distributed as the boundary wall expands.