In Catholicism, at least, both second "marriages" are invalid and are adulterous, as the "civil divorce" cannot actually dissolve the original marriage. CCC paragraph 2384 reads:
Divorce is a grave offense against the natural law. It claims to break
the contract, to which the spouses freely consented, to live with each
other till death. Divorce does injury to the covenant of salvation, of
which sacramental marriage is the sign. Contracting a new union, even
if it is recognized by civil law, adds to the gravity of the rupture:
the remarried spouse is then in a situation of public and permanent
If a husband, separated from his wife, approaches another woman, he is an adulterer because he makes that woman commit adultery, and
the woman who lives with him is an adulteress, because she has drawn
another's husband to herself. (St. Basil, Moralia 73,1:PG 31,849-852.)
The spouse who was abandoned may not be guilty of the divorce, but would be guilty of adultery if (s)he entered into a new "marriage" while the errant spouse was still alive. CCC 2386 reads:
It can happen that one of the spouses is the innocent victim of a
divorce decreed by civil law; this spouse therefore has not
contravened the moral law. There is a considerable difference between
a spouse who has sincerely tried to be faithful to the sacrament of
marriage and is unjustly abandoned, and one who through his own grave
fault destroys a canonically valid marriage.
It may seem "unfair" that the innocent spouse doesn't get to have the "fun" of cheating, but that's like the prodigal son's brother's complaint. It's a mistake to think that the sinner is having some sort of fun that the innocent are left out of. In reality, it's better (for one's own happiness in the long run) to be virtuous even if that means some short term suffering.
In these modern times, a spouse who has been abandoned will often seek an annulment from the Church (a decree of nullity) that claims the original marriage was never valid. With such a decree in hand, he or she can marry again. The issuing of annulments is probably overused, and is often cynically described as "Catholic divorce" because of how carelessly some bishops seem willing to grant them, but that is the state of things today.