AVS gets my vote with a single caveat: the free version places a
"logo" for the entirety of the video when you chose to "produce" (export into one of the common, non-proprietary formats):

As far as I can tell, you cannot choose to have the AVS logo placed in any other location than the middle of the image and it is not transparent.
Assuming this is not a deal breaker for others, continue reading...
This software is not "professional" grade, but it is EASY to use and if your needs are simple, it could be the answer for you as it looks to be for me!
My personal need: I coach a volleyball team. I want to present video to the team where I can pause and highlight where players should be in a given moment as well as show where the path of the ball is likely to be. So, I created a few simple graphics (in a different application) - a triangle to be used to show the possible path of the ball, a circle and a straight arrow.
I loaded these graphics into AVS and I was ready to go! AVS has individual channels for "special effects" (built in), graphic files, audio files, text, and voice recording for voice over! To get more than one of a given type on the screen at the same time, just right click the given channel and click the "Add line" option.
I dragged two cones to show the possible path of a hard driven ball when a blocker is in place and two circles to show where the defensive player should be within the area of each triangle (so 4 image overlay channels are used).
AVS allows editing the image (right-click on the image where it is placed in a timeline)! You can resize the image, rotate it, change its transparency and add a few built-in filters which also have their own transparency settings, change the duration that the image with appear and set image transitions. Perfect for my simple needs! [Note: if you create a .png image with the transparency you want before importing the image, you will not have to touch transparency within AVS]
I have only played with AVS for about 15 minutes, and it is the first of 4 free applications that I have test driven, but I am happy enough that I suspect I will end up paying the $40 to be able to produce finished product without the logo after reviewing the other 3! Bravo!
Should AVS be "listening", I'd suggest allowing the user to move the logo to one of the 4 corners in addition to the center and (if the important people in the company allow it) to make the logo partially transparent!
Edit:
VSDC Freeware
This is much closer to a full fledged video editor than is AVS and it is available as a completely free product to boot. But its power comes with a steeper learning curve which moves it beyond what I, like the original poster, am looking for. Negatives for me... 1) unable to rotate images I imported. If it is possible, it wasn't obvious to me how. 2) Unable to draw triangles (to get the cone effect of where the ball could be expected to travel (that would negate the need for me to rotate imported images, at least). 3) I didn't find the ability to change line thickness, but I am guessing this is due to me missing that setting somewhere.
I also found a couple of lightweight products with very few options, very little to learn! They allow screen shots - including video capture of whatever is on the screen. So put video full screen and capture away! Cool because? They also have very basic drawing tools. So it is possible to draw directly on top of the video. So in a live setting, you could pause a video and scribble, doodle, etc an overlay. Supposedly this can be captured while capturing video though I only verified the screen capture (.png file). Anyway, the two products I test drove:
Ink2Go: http://www.ink2go.org/download/win Trialware
Basic shapes: but once placed cannot be selected, moved or rotated - and except for the arrow/line can only be placed square with the frame (no acute angles). No transparency. Does place a nice arrow that can be placed at any angle. Shaped while not transparent can be filled or outlined.
Epic Pen: https://epic-pen.com/ Freeware
Sadly, this lacks the arrow drawing capability of Ink2Go. But I find it superior in every other way to Ink2Go (which I have already deinstalled). Objects drawn can be selected, moved, rotated!, and given transparency. Also the stroke size spans a larger range than does Ink2Go.
I see myself using AVS and Epic Pen as I move forward. If I ever need anything more than quick, easy markup for video and am willing to undertake the learning curve I will revisit VSDC.