Garrapata wrote:

...I cut /chop them up as much as I can and put them in the compost, the
ones that don't break down completely are sifted out and put back in the
next batch...or I don't sift them out and spread them with the compost.
Many of my beds are woodland-natural so larger pieces blend in and look
like they belong.

The sticks start to get soft over the winter in the compost pile if
they're not really thick. If you rototill your compost in, the tiller
will break some of the sticks up in the process. They will compost in
the soil just fine.

The process of composting requires nitrogen (green stuff) and carbon
(brown stuff). Putting too much carbon (e.g. dead sticks) in will tie up
the nitrogen (temporarily) while the dead sticks (carbon) break down.
Once the sticks are composted the nitrogen will be available again. The
speed of the process depends on the surface area of the carboniferous
component. Sticks will tie up some of the nitrogen, but something really
fine like sawdust will tie up much more of the nitrogen and may inhibit
plant growth. For this reason you want to make sure that animal manure
containing fine bedding such as wood shavings or sawdust gets well
composted before applying it to your garden. Sticks are generally not a
problem unless there is nothing else in your "compost".