When using dual monitors, is it possible to get the screenshots to appear on the desktop of the screen it captured?

I'm not using a mirrored display, I connect my monitor to my laptop to do my main work and use my laptop for keeping track of queues and emails. Throughout my day I take many snips/screenshots and I always have to slide over to a clear desktop on my laptop to grab and drag the new screenshot to the desktop on my monitor. The screenshots always stay afterwards on my monitor desktop.

Is there a way to there a way to change the screenshot default so that it appears on my external monitor desktop when I snap something on it?

Or maybe change the default "desktop" to be my monitor?

Files sent to Desktop will always land on the right side of the primary display - under your disk drives if you have them showing.

You can change which screen is primary, & though I can't test as I don't have a laptop, the primary is very likely to reset to internal every time the external is unplugged.

Change primary by dragging the white menu bar segment as shown below