As well as Andy's answer above, they have another property in common with the full braid groups $B_n$: they satisfy homological stability. There are two ways (that I know of) that one can prove this. The first is in the paper:

M. A. Guest, A. Kozlowsky, K. Yamaguchi, Homological stability of oriented configuration spaces, J. Math. Kyoto Univ. 36 (1996), no. 4, 809--814.

A sketch of the argument is as follows. It's enough to consider rational coefficients and $\mathbb{F}_p$ coefficients, for each prime $p$, separately. With $\mathbb{F}_2$ coefficients $E_n$ automatically inherits homological stability from $B_n$; this is actually true for any family of index-2 subgroups, or more generally double covering spaces, by using the mod-2 Gysin sequence and considering the double covers as 0-sphere bundles. (This only works for mod-2 coefficients since in general there is no Gysin sequence for 0-sphere bundles.)

When $F$ is a field of characteristic not 2, the homology $H_k(E_n;F)$ splits as
$$
H_k(E_n;F) \cong H_k(B_n;F) \oplus H_k(B_n;F^{(-1)})
$$
where $F^{(-1)}$ is the local coefficient system where the odd braids act by multiplication by $-1$. A model of the classifying space of $B_n$ is the configuration space $C_n(\mathbb{R}^2)$ of $n$ unordered points on the plane. There is a result of Boedigheimer, Cohen, Milgram and Taylor which calculates $H_k(C_n(M);F^{(-1)})$ in the range $k<\mathrm{dim}(M)n$ for any even-dimensional manifold $M$, and the above paper uses this to compute the $H_k(B_n;F^{(-1)})$ summand in this range. It turns out to be zero in a stable range, and so homological stability for $E_n$ follows from homological stability for $B_n$.

The calculations of the Guest-Kozlowsky-Yamaguchi paper actually work more generally, for configuration spaces on any open connected surface, so homological stability is also true for ``alternating *surface* braid groups''.

The stable range is worse than that for the full braid groups, however: $H_k(B_n;\mathbb{Z})$ is independent of $n$ for approximately $n\geq 2k$, whereas $H_k(E_n;\mathbb{Z})$ only becomes independent of $n$ for approximately $n\geq 3k$. The obstruction to $E_n$ having the better range lies entirely in the 3-torsion.

The second way I know of proving homological stability for the alternating braid groups is kind of a shameless plug, as it's something that I wrote (apologies if this is inappropriate; this is my first MO answer). It works more generally for what could be called ``alternating configuration spaces'' (but which are actually called oriented configuration spaces since that's what they were called by the GKY paper above). It's in the preprint

Martin Palmer, Homological stability for oriented configuration spaces, arXiv:1106.4540.

and uses a method of ``taking resolutions'' adapted from that of

Oscar Randal-Williams, Resolutions of moduli spaces and homological stability, arXiv:0909.4278.

Just for historical completeness, I should also mention that homological stability for the alternating groups $A_n$ (which can be thought of as the alternating braid groups on $\mathbb{R}^\infty$ if one is so inclined) was proved much earlier, as Proposition A on page 130 of the paper:

J.-C. Hausmann, Manifolds with a given homology and fundamental group, Comment. Math. Helv. 53 (1978), no. 1, 113--134.

using the same kind of decomposition as in the GKY paper.