I think it depends on what you and the client have worked out. Social Media management is my primarily job for some of my clients-this covers every single task. When we first go over objectives they let me know what they see in their FB/twitter accounts and I tend to cover the work from there. If there is an especially personal post/comment or question that the small business owner should be involved in I do save it and let them respond but when I cover the social media/PR
I do so completely, often it is a very daunting task for some without the experience of social media in the first place.
If you are taking on FB/Twitter posts for a very specific small business it may be harder to take on the posts on your own w/o excessive co knowledge, maybe best leaving details to the owner. An educational software company I created FB/twitter for is large enough the CEO didnt have time to touch social media so it's been a great learning experience on how to cover everything start to finish.
When you do upkeep but not posts are you just covering image uploads/box or tab management? Is there a lot for you to do after the initial setup? I'm interested to hear how others "do social media"