To get transparency working with other objects in eevee, you have to tick a couple not-so-obvious boxes.
First, in Render Settings, tick Screen Space Reflections:
And then under screen space reflections, tick Refraction.
Then in the material menu of the transparent object, tick Screen Space Reflections:
Then adjust the refraction depths slider just below it. The default is 0m, which gives some really extreme refractions that doesn't look right. (EDIT: or maybe it does? Spheres are suppose to have inverted reflections)
The lower refraction depths is (excluding 0), the less refraction you get, so 0.0001m basically mean no refraction.
With a refraction depths of 1m, I get this:
There are still a few caveats though, mainly that (AFAIK) you can't have 2 objects that both have materials with SSR checked behind one another. The one behind will just not show up. You can however have a transparent object without SSR checked behind one with SSR. So this only matters if you have 3 or more objects in a row.
With refraction depths set to 0.0001,
Middle sphere without SSR:
Middle sphere with SSR: