Theoretically, it should follow, i.e. If you post on DW as public, it should post on LJ as public If you post on DW as access-only, it should post on LJ as friends-only If you post on DW as private, it should post on LJ as private
Exceptions: If your default on LJ is friends-only, and you post on DW as public, it should post on LJ as friends-only. If you post to a custom access filter on DW, it will post to LJ as friends only, i.e. including friends who are not on that filter. (This may change - the crossposter is currently in alpha.)
Giving access to people on DW (including OpenID users) does not affect your LJ friends list, so does not affect who can see entries on LJ.
Crosspost filters
If you post on DW as public, it should post on LJ as public
If you post on DW as access-only, it should post on LJ as friends-only
If you post on DW as private, it should post on LJ as private
Exceptions:
If your default on LJ is friends-only, and you post on DW as public, it should post on LJ as friends-only.
If you post to a custom access filter on DW, it will post to LJ as friends only, i.e. including friends who are not on that filter. (This may change - the crossposter is currently in alpha.)
Giving access to people on DW (including OpenID users) does not affect your LJ friends list, so does not affect who can see entries on LJ.