Is it possible to use puree in a siphon whip

Yes, but chestnut purée comes in sweetened and unsweetened varieties, so making your own depends on what you'll be using it for. Chestnuts are pretty versatile, but I've come across using sweetened purée in old world desserts, while the unsweetened purée is typically used with root vegetables and winter squashes (the European variety of chestnut drop in late fall).

You'll start by scoring the skin of the raw nuts and roasting them at 400° (this prevents them from bursting). After this, peeling them is easy work. You will then boil them, optionally in a sugar syrup. Finally, reserve the liquid and puree in a food processor, adding back the liquid for the desired consistency.

About a pound of chestnuts per cup of purée.

Raspberries aren't just seeds and juice, though they're certainly not as fleshy as some fruits. The difference between apple puree (apple sauce) and apple juice is much more obvious. In any case, you should be fine forcing them through a sieve as you suggest.