If I am eating the sub at a sit-down restaurant or at home, saucy is fine. It reminds me of when I was younger and my mom would be cooking pasta and other stuff for dinner, and to shut me up in the mean time, she would put some sauce on bread for me to snack on until dinner was ready.
But if I am eating one at work, or in my car, or in a more formal setting - it would drive me nuts if it was swimming in sauce like that. The last thing I want is to be wearing half the sauce on my shirt by the time I am done, not to mention having sauce all over my face looking like a pig eating it. I also wouldn't want it sauced up so much, if I had a longer drive back home before I could eat it. I never thought to ask for sauce on the side though, that's a good idea.
Subway drives me nuts though with how they make their subs. I hate it when I ask for a little bit of mayo on something and they spray like the equivalent of a whole jar of mayo on it. As chincy as they are with their meats and cheese, you would think they would be equally cheap with their condiments. For some reason, they also smash the crap out of their subs when cutting them in half.