Hey Ken...
Every transplant program is different. In some smaller hospitals, not only is a urinary catheter required, they make YOU put it in yourself! Most of the larger centers will do this for you once you're under, just pray you don't have some beastly, man-hating nurse jamming it in you....while you're still awake! The last thing you need is some ex-olympic Russian weightlifter with rough, 'wheat-farmer' hands manhandling your catheter into place! Also, make sure they use the newer flexible catheters as well, the old ones are metal, they rust easily, they're rigid and have sharp barbs on them to hold them in place.
Just kidding. In all honesty, I had the same concerns and scary thoughts concerning catheters before my donations as you do. You can relax though, they put it in while you're out. You'll wake up with it attached to you and frankly....it not only looks pretty cool, it takes the hassle and strain of peeing away for you! (Ain't nobody got time for that!) Made me feel like a super-hero! During my first night of semi-awareness, I was the invincible 'Pee-Man"! I even took pictures of it! The next morning when they took it out, I was barely conscious of the fact they were removing it..it just felt weird...but no pain at all! Trust me, the catheter is the least of your concerns. You're doing a very noble and generous thing here!
Be well...
Chandler