I live in the area you’re looking at. We love it. The weather is awesome and massively underrated as a plus point IMVHO. Lake all summer, skiing all winter.
Local or international depends on your longer-term plans (which I know are hard to know!) and contractual set up.
If you’re on an expat contract and likely to move on within a 5 year period, and DH’s work is paying for schooling, I’d go international. Local school would be too much unnecessary disruption.
If it’s a local contract and with little or no financial help I’d go local. Unless DH will be on CHF 250-300k+ in which case you can afford it yourselves. The big international schools will be CHF 35k per year per child. There are smaller private bilingual schools for around CHF 20k but usually offer a Swiss curriculum (so you may as well save the cash and go local).
We’re on a local contract for my work. My kids have been in local school for all of primary and are very happy. But we have seen many families struggle once kids older than about 7 start with the local system. Picking up French will take time. None of this fluent in 3 months rubbish!!! They sound fluent as they get the accent straightaway but I would say allow 2 years for full fluency. Another poster mentioned the critical part of schooling at the end of primary which is very true. It’s a bit like a compulsory 11+ which then streams them for middle school. Deffo read up on this given the age of your eldest.
Many new arrivals also get frustrated with the fact that kids go home for lunch in local schools. You can pay for lunch at school but places are very limited and you are lower on the priority list if only one of you works. We pay nearly CHF 600 per month for lunch (no before/after school) 4 days per week (Wednesday afternoons all schools are off).
In the international schools even if in a bilingual stream do not expect your kids to reach full fluency in French. Kids are lazy buggers when it comes to languages so will default to English as much as poss. Even if they do half their time in French, they’ll be pretty good, but as it’s not as immersive as local school, it’ll take longer to acquire the high levels. But perhaps that doesn’t matter too much - depends on your longer term plans.
Anyway, I guess my point is do the research re schools and you’ll be fine. It’s a beautiful corner of the world and a great place to bring up kids.
Oh last comment. Do make sure it’s a decent salary. It’s an expensive corner of the world too. Fine if you’re earning ok but do your sums first too. Allow CHF 5-5.5k for rent + health insurance.
Enjoy! And let us know if you have other questions!