Glyph of Smite gives a 20% increase to Smites cast on targets burning with Holy Fire.
Does this Glyph give its 20% bonus to my Smite when another priest has Holy Fire ticking on the target (and I do not)?
It is only the presence of your Holy Fire that determines whether or not the glyph's bonus damage applies or not, similar to any number of other symbiotic spells in WoW.
That said though, Holy Fire has a higher damage per cast time than Smite, so there's no reason why you'd ever cast Smite when Holy Fire is off cooldown and its DoT is off, making your question kind of moot.
Some numbers to show you what I mean visually: