If you want to integrate with the Swiss and need to feel part of the community, your only choice is to send them to local school if the children are young enough. It's not like living in London or New York etc.., where lots of kids go to private school and it doesn't really matter, because very few Swiss families -- even the rich ones -- choose this option. Your children simply won't be accepted by the local community unless you take this step. I can speak only for the German side, but if your kids can't communicate in Swiss German, making local friends will be pretty much impossible, regardless of how many football or scouts classes you enroll them in. Their social circle will be limited to kids who go to their school, and they are unlikely to live nearby.
Obviously, there are exceptions, and probably some people have made it work but I, for one, would be exhausted by all the effort. Frankly, the Swiss are tough nuts to crack at the best of times, and it has taken us this long just to get invited to birthday parties etc...
And yes, it will require at least one parent to learn some form of German and to try to integrate, as difficult as it sometimes is. And it will require you to get a handle on how the Swiss educational system operates and yes, it is very different from the Anglo-Saxon system and yes, being the foreign kid is tough educationally and socially. But if you're not willing to make that effort, your child is unlikely to feel accepted because other parents will shun you. It's not fair, but there you go. What you put into it is what you get out.
If you don't care about getting to know the Swiss and don't care if your kids have local friends, then international school is an option. But at this point, you really have to ask yourself if a permanent or semi-permanent move to Switzerland is really such a great idea.
Obviously, if you're only going to be here for a couple of years, and you can afford the fees, don't think twice about international school. Local school is simply not worth the gargantuan effort, and by all accounts, the international schools are warm, nuturing environments.
A couple of caveats: 1) if you both plan on working full time and send them to local school, you will have a hell of a time trying to stop up all the holes in the schedule. Swiss schools pretty much rely on one parent being available a good part of the time. 2) if your children are older than 10, they had better be pretty motivated. The pressure and the streaming starts early. Not sure if my kids could have handled it.
Frankly, my biggest issue -- and one that I would really like someone with older children to write about --is what happens when they hit high school? Only a quarter of Swiss-educated kids make it to University and I'm wondering if it is possible for children whose parents are non-native speakers to make it that far. It is very difficult to get a good job outside of Switzerland without at least a BA or BSc. What then? Will their English really be good enough for US/Canada/UK?
Good luck to all of you making this choice