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. https://www.dropbox.com/s/v83svmryxawhpd4/Libre%20Office%20Issues.mp4?dl=0
From screen cast, OP is on OS X 10.9.4 using LO 5.0.3.2 Setting as duplicate of bug 81759 LibreOffice replaced NSTextInput with NSTextInputClient with http://cgit.freedesktop.org/libreoffice/core/commit/?id=179334dd45828d02e4e323cbdda435963fa3f4a5 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.