You’ll have to dive into the delights of Page style because header/footer is a property of page styles.
Open the style panel with F11 and click on the page icon (fourth from left).
To preserve the possibility to have un-headed chapters, create a new style: right-click on Default Style and choose New …. Give it name Chapter (or any other to your liking).
Tune the general lay-out in Page tab.
Got to Header tab. Check Header on, uncheck Same content left/ight.
Click OK to record your page style definition.
Modify your Heading 1 paragraph style definition (I suppose all your chapters begin with such a paragraph containing the title). In Test Flow tab, check Insert and With Page Style in Breaks section and choose Chapter style in the drop-down menu. Click OK.
With this modification, page style is automatically set to Chapter whenever you start a new chapter.
Now, click in the header area of an odd-numbered page. Insert → Fields → Other.... Go to Cross References tab if not already there. Select Headings in the left pane, Reference in the bottom pane and the appropriate chapter in the top pane. Click OK.
Set your header paragraph to Header right so that you can apply a different alignment (and other attributes) for left/right pages.
Do the same in an even-numbered page with style Header left.
Of course, this is very short to start with. Experiment with all the endless possibilities to fulfil your needs.
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