Sometimes the tensioner pulley is needed to lift the chain over the chainstay because a straight line between the bottom of the sprockets would collide with the chainstay.
Some cruiser type bikes have the chainstays arranged like a U and these are most likely to work (and have horizontal dropouts) but mountain bikes and hybrid/road bikes may be more of a V or trumpet shape and so they need the chain tensioner to lift the chain until it gets to the wider part.
You can't have both chains with no tensioner on a single speed bike because the wheel wouldn't be straight when both chains are at correct tension, or vice versa.
It is probably better to have a pulley whatsit chain tensioner on the pedal side and have the engine side with no tensioner, if the frame allows this.