I've been told numerous times that Apple's dictation works with LibreOffice's new version 5.3. Here's a video showing that it is not working for me.
From screen cast, OP is on OS X 10.9.4 using LO 126.96.36.199
Setting as duplicate of bug 81759
LibreOffice replaced NSTextInput with NSTextInputClient with
NSTextInput was deprecated at OS X 10.6, and while there is likely more to be done numerous user report the NSTextInputClient based support works for them--except on OS X 10.9 builds.
Unfortunately, for OP it is probably past time to update OS X version.
OP had previously posted issue as bug 93035 for LO 4.4.4, that also was dupped to tdf#81759
*** This bug has been marked as a duplicate of bug 81759 ***
So, what you're saying is i need to update my OS in order for the feature to work?
While the LO implementation of NSTextInputClient probably needs improvement/completion, the issue is on the OS X side--assuming the findings of 81759 hold true.
So since Apple will not do anything for Mavericks 10.9.5, your best bet is to look at 10.11.1 El Capitan--so, yes.
You need to update.