Security of Fast Two-Party ECDSA Signing The paper "Fast Secure Two-Party ECDSA Signing" by Yehuda Lindell describes a system in which two parties, $P_1$ (with 1/2 of a share of a long-term ECDSA signing key $x_1$) and $P_2$ (with the other 1/2 share, $x_2$), cooperate in order to produce an ECDSA signature, $P_1$ being the party that actually outputs the final signature.

In Section 3.2, describing the process for distributed signing, it is said that a hypothetical attack by $P_2$ implies sending a malicious message such that if $x_1 \bmod 2 = 0$ then the final signature is incorrect (and hence $P_1$ will not output a signature), and if $x_1 \bmod 2 = 1$, then the final signature is correct (and hence $P_1$ will output a signature). The author refers then to the security proof for details on how to handle this.

If I am not mistaken, it is in Section 4.1, pages 20-22, where this is addressed (there is an alternative security proof in Section 5, but I am focusing on the one in Section 4). In this section, security is proven by reducing to the security of ECDSA (i.e., if we have an attacker $A$ that breaks the 2-party ECDSA, then we can build an attacker that breaks ECDSA). 
Specifically, for the matter at hand, the strategy is to have the simulator simulate $P_1$ aborting at some random point. This makes the simulated view be indistinguishable to the real view with non-negligible probability, so we can use $A$ to break the 2-party ECDSA, then use this to break ECDSA, and so on.

Now, my doubt is: how does this demonstrate that a malicious $P_2$ is prevented from learning the LSB of $x_1$ if the hypothetical attack mentioned in Section 3.2 were possible. Or does this imply that, in practice, we would actually need to force $P_1$ to abort randomly? In that case, is this realistic in practice?
In real life, $P_1$ of course does not abort randomly; this would make no sense. The idea is as follows. Assume that a cheating $P_2$ can forge a signature with probability $\epsilon$ when interacting to obtain $p(n)$ signatures are signed with the system. Now, if $P_2$ cheats then there is a first place where $P_1$ aborts. An adversary for ECDSA can simulate everything and guess where this first abort happens and will be correct with probability $1/p(n)$. This implies that the adversary for ECDSA can forge a signature with probability $\epsilon/p(n)$. Since ECDSA is assumed to be secure, this means that $\epsilon$ must be negligible.