Sorry guys, but there is no koshi nage in judo. Many of the judo throws use koshi (hip throws) while others use feet, legs, shoulders...etc.
I didn't do judo for over 7 years now, so my memory is not good, but throws like ippon seoi nage, morote seoi nage, o goshi, uki goshi, harai goshi, hanae goshi...etc. are all hip throws, and they have nothing to do with our koshi nage.
The main difference I can see (which is a big difference by the way) is that aikido's koshi nage is executed at a 90 degree angle betweem nage and uke feet, while in judo, uke and tori feet and hips are usually parallel or almost parallel.
In judo, you can throw an unwilling person using a hip throw (even if he's pulling backwards, while in aikido nage should be using uke's movement otherwise it would be more difficult to execute.
Well... to be pedantic, koshi nage merely means hip throw, so, there are several koshi nage "in" judo. They're not CALLED "koshi nage", but are named according to how they're executed. O Goshi is sort of big hip throw, hane goshi was translated as spring hip when I practiced judo many years ago. Tsuri-komi-goshi - I forget, but it's a bit like ryote-dori-koshinage. Uchimata - inner spring hip, or something like that.
The "seoi nage" throws (ippon seoi and morote seoi) were classed as "hand" throws or "shoulder" throws...
The training in judo does, as Edward remarks, have the hips parallel with the feet generally inside the line of uke's feet. One of the differences between the "finish" of judo throws and that of aikido throws is due to the points system in judo - if you release uke to fall on his or her own in judo, they will do whatever they can to avoid landing on their back and giving up ippon. So - training (well, my training) for judo was to slam uke to the mat as if we were trying to put them through the floor, and to stay with them to make sure they didn't twist out at the last moment...
Aikido seems to project uke towards the floor and leave him/her to sort out a safe landing - with the option of having a finishing "crunch" to help the person decide to stay on the ground instead of getting up to continue a "situation"...