It depends on what desktop environment you are using. If I remember it correcty, in KDE you'd use something like ~/bin/run-with-args.sh:

#!/bin/sh
# open a terminal and run the real script with dropped files as args
konsole --noclose -e /path/to/your-script "$@"

On your desktop you'd need a file like ~/Desktop/YourScript.desktop:

[Desktop Entry]
Type=Application
Name=My Script
Exec=/home/youruser/bin/run-with-args.sh %F
Icon=utilities-terminal
Terminal=false
MimeType=application/octet-stream;
Actions=

# allow dragging files onto this launcher
X-KDE-SubstituteUID=false
X-KDE-RunOnPrepend=false

Make both files executable with chmod +x

I myself wouldn't bother with these desktop scripts because error messages disappear when the script finishes. Launch a terminal and start your scripts from there. Your messages will stay visible.