yes, gravity waves are fluctuations in spacetime, we (according to theory) receive them as polarized plane waves which means they're directional .. so as you say, the 2 arms of the antenna will contract and expand differently, so that should disturb the equliibrium pattern of the 2 crossed lasers in a clearly measurable way (a clear 'wave-signature' which can be correlated to a specific astronomical event)

Here's a good explanation -

The UK's Dr Harry Ward, an international partner on Ligo, explained: "The effect of a gravitational wave - if you imagine it propagating towards you - would be to lengthen space between your head and feet, and contract your width. It distorts you in that way.

"So the way we try to detect gravitational waves is to lay out, L-shape, a set of mirrors. The gravity waves would shorten one arm between two mirrors while simultaneously lengthening the other arm between two mirrors."

.. pretty clear and simple .. however you mention something which I've also been wondering about -

Ears evolved to detect external differential compression of air, relative to an internal standard. A transient change in the "compression" of spacetime simultaneously effects everything nearby: the ear, the portion of the brain processing the signal, the internal and external air pressure, et cetera -- with the result that there is no local differential to detect.

.. hence the L-shape configuration I suppose - it's the relative change between the 2 beams which counts, this then gives the disturbance in the laser interference pattern as you say.

The earlier, less sensitive detectors were resonant bars made of ultra- resonant materials like niobium:

Niobe was last operated in 2001. It achieved a noise temperature of less than half a microdegree Kelvin, corresponding to a gravitational wave strain sensitivity for short bursts of about 3. 10-19. This sensitivity is sufficient to detect supernovae, neutron star births and coalescences of neutron stars and black holes in our galaxy.

.. and they are still being built today, eg in the Netherlands:

The MiniGRAIL detector is a cryogenic 68 cm diameter spherical gravitational wave antenna made of CuAl(6%) alloy with a mass of 1400 Kg, a resonance frequency of 2.9 kHz and a bandwidth around 230 Hz, possibly higher. The quantum-limited strain sensitivity dL/L would be ~4x10-21. The antenna will operate at a temperature of 20 mK. An other similar detector is being built in São Paulo, which will strongly increase the chances of detection by looking at coincidences. The sources we are aiming at are for instance, non-axisymmetric instabilities in rotating single and binary neutron stars, small black-hole or neutron-star mergers etc.

.. so the idea here, is to have a mass which is ultrasensitive, ie tuned, to a given narrow frequency bandwidth. The frequency is chosen from theoretical GR predictions for various massive or dynamic events; the tuning of the mass is designed by material, shape, suspension etc. The Niobium detector was so sensitive at its resonant frequency that it had a ring-down time of about 6 months .. the time taken for it to stop ringing after being excited once.

These earlier detectors are better examples of the 'paradox' you mention - wouldn't everything be compressed, so we wouldn't detect any change .. the bar would compress & expand along with us and everything else, so how would we notice anything? - the way to see it here is to see the gravity waves as (sort of) impulses, after all, like all waves they are a transference of energy - here, gravitational energy. So when one of these 'impulses' hits the resonant bar, it will ring, as if you had hit it with an external acoustic impulse. The difference is these gravity waves aren't external to the mass, but happen to the space it occupies. That's a mind-bender but in the end means just that the gravitational energy (of a given frequency) will be transferred to the bar and set it ringing at that frequency.

But to be honest, IMO the issues must be more subtle, and the real question is (in physics-speak) 'how does space, and hence gravitational wave energy, couple to matter'. Obviously that must have been dealt with and answered, but like you I'm hoping to find out more.