We did this for two years, and it was a great arrangement. You've gotten great advice, I would just add:
We gave our nanny 3 weeks off, paid, and 5 sick days per year. We were paying her well, but not super well, and this was a benefit we could do to sweeten the job. It means you need to arrange ahead of time when exactly the nanny will have off (vacation, national holidays, etc), and what to do if she goes over her sick time (in our case, making up hours).
Make sure you're on the same page re discipline, and check in often with the nanny and the other family along the way. When they're 1 it won't matter so much, but what happens the first time (or 8th time) one kid bites the other?
Make sure you know how long an arrangement you're committed for. We said we'd do a year, then decide whether to opt for a second year. Despite this, the first family we partnered with changed their minds a couple months in and pulled out.
Lucky for us, we found another family we liked much better.
It was a stressful couple of weeks, though.
Re sickness: how sick does the kid have to be before you don't send them?
Re payment: We guaranteed a certain number of hours of pay/week that we were each responsible for (for us, it was like 37). So if a kid was sick, we'd still pay the nanny up to that number of hours. Then we figured out week-by-week who used which hours. We paid a joint, higher rate for time when the nanny had both kids (the 37 minimum and whatever other joint hours we used), and a slightly lower rate when she only had 1 kid (eg, one of us wanted an extra hour of work that day). We also paid overtime over 40 hours/week. Our nanny had keys to both places, so, for instance, she could walk child #2 back from child #1's house if child #2 was staying later that day. I HIGHLY recommend using an Excel spreadsheet to track hours etc.
Re location and food: we switched weeks back and forth, which worked out well. With the first family, we said the family hosting would provide food for the kids and nanny that week, which was awesome. With the second, our kids had different allergies, so we ended up sending lunch for the kids (not nearly as convenient). If you end up feeding each other's kids, you should make sure you're on the same page about what they're eating.
Re money: we each provided the nanny with some petty cash (to buy the occasional bagel with the kids), and she let us know when she needed more. It was only an occasional thing.
Also a double stroller is nice, but so is putting one kid on your back and strolling the other. That's what our nanny preferred, since it was easier to get around on city streets.
