How much overlap will the Andromeda galaxy and the Milky Way have when they collide? Measurements of Andromeda's blue shift let us conclude that the distance between the Andromeda galaxy and the Milky Way is decreasing and in a few billion years they will "collide".

The blue shift only yields the radial component of Andromeda's velocity vector. It is my understanding that measuring the tangential component is crucial in determining whether a "collision" will actually happen (in a gravitationally bound two body system, for point-like bodies to collide, the relative velocity component must point exactly towards the other body, i.e. the tangential component must vanish).

Now, galaxies are not point-like, so some small nonzero tangential component might lead to a collision where at least some galaxy arms intersect.

Has the tangential velocity been measured? If so, how? How central is the collision (bulge into bulge, bulge into arms, arms into arms)?
This page, from June 2012, contains a fairly detailed summary of three papers that considered the tangential velocity and collision problem.  I'll quote a few choice bits from that page:

Andromeda (M31) and the Milky Way are the two largest galaxies in the small group of galaxies called the Local Group. At the moment, M31 is about 770 kpc away from our galaxy, but Doppler measurements of the line-of-sight velocites of its stars have long shown that it is inexorably falling toward us. The exact nature of the collision, however, has so far remained unkwnon. Will it be a miss, glancing blow, or head-on smashup? This depends on the proper motion (sideways motion) of Andromeda in the sky, which is extremely hard to measure.

At last, the authors of this study manage to collect extraordinarily accurate Hubble Space Telescope observations of the tangential motion of M31 over a five-to seven-year period, and these data remove any doubt that Andromeda is destined to strike and merge with our Milky Way in four billion years. The Triangulum galaxy M33, the third most massive galaxy of the Local group, which is bound to M31, will likely join in the cosmic boom (with a small chance that M33 will hit the Milky Way first).


The final result is that M31 has a radial velocity with respect to the Milky Way of about  -109 km/s, and a tangential velocity of about 17.0 km/s (&lt;34.3 km/s at 1-sigma confidence). These numbers clearly imply that the velocity vector of Andromeda Galaxy is statistically consistent with a radial (head-on collision) orbit towards the Milky Way. This means that the Milky Way-M31 system is bound, and that the two galaxies will merge, as investigated further in Paper III.



The simulations show that the Milky Way and the Andromeda galaxy will merge, consistently with the radial (head-on collision) orbit deduced in Paper II for M31, and  that the first “pericenter” (closest approach) will occur at around 4 Gyr from now. For our Milky Way,  the encounter has 72.2% probability of being prograde (the galaxy spins in the same direction as the flyby). In 41.0% of the Monte-Carlo orbits M31 makes a direct hit with our Milky Way, where the authors define a “direct hit” as an encounter with a first pericenter distance less than 25 kpc. The two galaxies will eventually merge after 5.9 Gyr, and the radial mass profile of the merger remnant will be significantly more extended than the original individual profiles. Roughly speaking, this profile will follow the $R^{1/4}$ law characteristic of elliptical galaxies, in agreement with the predictions from the numerical simulations of major mergers of spiral galaxies.

I'll note that $25 \text{ kpc} = 25,000 \text{ pc} \cong 81,540 \text{ ly}$, and that the Milky Way is only about 100,000 ly in diameter, so the "direct hit" definition is quite a substantial distance here.