It would have to be "... on their own ground".

But the expression I know is the above.

I second Phillip

I'll third that...
cheers!

I can beat someone on his own ground but with my weapons!

I think this could be better????
To defeat someone with his own weapons!

Yes, you can Steve!

noch ne Variante
on his/her own turf

As Steve53 points out though, "on his/her/their own ground/turf" doesn't have the meaning of "with their own weapons/at their own game".

I'd say it is a different image, but the same meaning.
Fighting somebodys game /on sb. turf means accepting his rules thus his weapons?!?

In general we know the meaning, if we get specfic we have to make a differece between:
same weapons - gleiche waffen
own weapons - eigene Waffen

wolfgang, no, I don't think the two expressions mean the same at all.

Take for example the sentence:

"Blair was beaten on his own ground (national security) with a wound in his most tender spot (trust)." [comment.independent.co.uk/columnists_a_l/simon_carr/article2486604.ece]

It doesn't mean Blair was beaten at his own game or with his own weapons, but in an area he was thoroughly conversant with, one that he had perhaps used to boost his party's standing, etc. The expression concerns the subject in which he was beaten. (The expression comes originally from the sporting world, I would guess: home ground, away ground.)

If someone beat Blair politically by means of lying, spreading dubious truths, manipulation of the media, etc., then you might say (with reference to Blair's supposed weapons of mass destruction, etc.) "Blair was beaten at his own game". I.e. this expression concerns the means which are used.