You will have two half-shadows, and maybe an overlapping part where no direct light reaches the ground.
Thus there will be a shadow where light of neither reaches the space. There will be a partial shadow where only light from one star reaches the space. And then there is of course the space which is illuminated by both.
You totally right: one can try out the result at home in the evening or night hours: use two direct light sources, like spot lights at different angles, dim them to their relative brightness as desired. Don't use ambient light which illuminate walls, ceiling etc. Then place an object such that those two light sources illuminate the object and observe the shadows. You can observe a similar effect at night when you walk or cycle a street illuminated by spaced-out street lights: you will see shadows of yourself in two different directions.
The closer together the two stars are in the sky the more the two shadows will overlap and the more the shadows will look like you know it from Earth. The more apart they are, the different the directions of the shadow will be.
The difference in relative brightness (or shadow-ness) depends directly on the relative brightness of the stars. Of course the indirect light from the atmosphere, thus the scattering behaviour of the atmosphere influences the difference between complete shadow (only indirect light) and illuminated area (indirect light + direct light), too. That might vary a bit from planet to planet, too. On Earth the difference of the brightness of shadowed area compared to directly illuminated area (according to wikipedia) is of the order of 20000 lux vs. 120000 lux (thus illuminated area is about 6x brighter than shadows), but less contrast when the sun approaches horizon or worse atmospheric conditions.