I have two HDRs made by myself using two different photographs.
My problem comes when I open both in Photoshop: I cannot copy paste one over the other so I can mix them together. How can I do that?
First of all, when making a lightprobe, the images need to be taken at 90-degrees from each other, not opposite sides of the sphere. If you take them 180-degrees from each other, the bits you want to replace will exactly overlap each other.
Secondly, you need to "unwrap" each image to a flat mapping that's a full 360 panorama--they aren't half-180-panoramas. The reason you can replace bits is because they're both almost-full panoramas. I'd probably use the commercial Flexify plugin for Photoshop to map mirrorball->equirectangular, although maybe the free Ornament plugin would work for this. Once you have 360-degree panoramas, you can use Hugin to rotate one of the images in yaw by 90 degrees, so they'll overlap exactly.
Then, you bring each image in on a separate layer, and using masks, you can combine the bits you want from the two images to eliminate the parts you don't want.
If you have both images open in different projects:
On the image you want to copy over another, right-click the image
layer and choose
Duplicate Layer. A dialog will appear where you
can select the document where you want to paste the layer.
You can try to drag & drop a layer from one project to another, but this doesn't always work.
If you want to open two (or more) images in the same document:
Load Files into Stack.