Which version of LW are you using? I am guessing you are referring to iTunes right?
You can share those iTunes playlists to play, but not to copy or burn. Not unless the files themselves are transferred to his iTunes library database.

For LW 5, Tools -> Options -> Advanced -> Files, & check option at bottom for share iTunes on LAN. However your husband will not see this playlist unless he is using iTunes at same time. (For LW 4, it's Tools -> Options -> iTunes ...)

Alternatives. A few I can think of. Various workarounds but a bit of mucking around & not sure which would be the most practical. One question is, where is your iTunes music library stored. Is it the default C:\Music or C:\Shared Documents\Shared Music or more likely C:\Documents & Settings\Account Name\My Documents\My Music\iTunes\iTunes Music
Reason I ask is playlists can be exported. But would only work if your husband had access to your iTunes music files ... if they are in your account then that might stop his access unless your account music folder is set as a shared folder via Admin control panel.

Then with the playlist, he can simply open it with iTunes using File menu -> Library -> Import Playlist

Birdy or someone else might have a good suggestion here.

If you are not referring to iTunes. Then you are able to save a list of songs as a playlist in LW 4 (not LW 5). However again, your husband would need access to the location on the computer to be able to access those song files.