A Detroit style locker will not fully lock unless there is torque being applied to the housing. With little torque application going around a corner one wheel is disengaged or ratchets to compensate for the speed difference.
This is also why it is not recommended to power through a turn on a solid surface with a Detroit style locker, powering through a corner can lock the axle and increase the turning radius significantly, as well as pushing the vehicle straight instead of turning.
This is the primary reason why limited slips and torque biasing diffs are used in on-road motorsports, and Detroits are primarily used offroad.
Personally I would install the locker on both axles, but for one I would install the detroit in the rear most axle, as torque is only being applied to the center axle if you select mode on the transmission control panel. Also installing it in the rear axles is much easier as you don't have to muscle with a 350 pound chunk, and the angle of the rear most axle really helps with install.
An automatic locker isn't recommended in the front for the reason mentioned above. For front axles in vehicles used on the street it's recommended to use a manual locker, clutch type diff or a torque biasing diff.