Here is an open-source alternative to Epic Pen. According to the author, it was inspired by Epic Pen and the purpose is to make it even easier to use, especially for presentations. You select from pens to draw things instead of changing individual settings of color, transparency and tip width every time, and each pen is a combination of these attributes and is configurable to your need.
Update: I may have found two options that aren't connected to the desktop but might serve your needs.
gInk: https://github.com/geovens/gink (open source)
PainterMouse: PainterMouse.com (freeware)
I've tested gInk and it works well, ideally with a stylus or touchscreen monitor.