To explain, I have a graphic with space between two objects. I want to make the two objects have a common boundary.

I have tried multiple path choices to no avail.

See the transparent background. I want that to disappear.

If it's for rendering on screen and not to use cnc for cutting matching pieces, the best option is to overlap one object by another on top.

Because even if the geometry matches, with the anti-aliasing you will still see through at the edges (known as gap issue).

From the detail in the screenshot it'd be as easy as deleting nodes in the blue path and raising the white one to the top.

If the design is for a cutter (rather than simply visual) a path-operation is usually used (sometimes via a duplicate as the "tool") or by using the shape-builder tool.

Please share an example Inkscape svg file for us to evaluate and demonstrate with.

"From the detail in the screenshot, it'd be as easy as deleting nodes in the blue path and raising the white one to the top."

Good answer. I have done that manually, but is there a way to do it with Inkscape?

There are several ways to edit nodes out of the path. Can rectangle select them, can add to the selection by holding the Shift key, or using the scroll wheel.

If it's about editing a very complex path -something like of the result of trace bitmap- then you can still break the path into its subpaths (Shift+Ctrl+K). 

As an in-between option you can split nodes at the start-end positions of those nodes you want to delete, 

break path apart, deselect the new subpath and combine the rest back together and connect the unconnected nodes.

Sometimes that option is faster, mostly it's less convenient.