In the way you have phrased the question, you have clearly stated "as it's needed" - if you know where it is needed, why would you show it in places it isn't needed?
Need is in the heart of the issue. A live chat offer is only useful when needed, when it's not, its an extraneous visual element.
A persistent live chat offer is often hard to ignore (depending on the visual design though). When such a feature exists, it is useful to make users aware of it. The homepage is a good place to put such offer - users can always come back to that page if they wish to chat. But a common knowledge in UX is that not all users land on the home page, many arrive from search engine into other pages.
First hand experience, with a system that had a small red rectangle at the bottom right corner, users has actively complained about it being there when they hardly need it. Nowadays you can easily reduce the footprint of such feature. Like showing a live chat rectangle when the users first enter the site, an animating it shortly after to a much smaller low-contrast icon.
But assuming the live chat feature is needed on each and every page is only valid on a few systems.
I think your main goals are:
- Make sure users are aware of the feature and where to find it.
- Minimise its visual footprint (size and how noticeable it is).
It is, after all, an on-demand feature: people should now where to find it, but not have it constantly in their face.