To the drafters of the 14th Amendment, the phrase "privileges or immunities" was synonymous with "basic civil rights." But in 1873, the court held that only some of the rights enumerated in the Bill of Rights restrict states by being "incorporated" into the 14th Amendment's "due process" clause.
Since 1897, the court has held, with no discernible principle, that some rights enumerated in the Bill of Rights are sufficiently fundamental to be "incorporated" but others are not. This doctrine bears the oxymoronic name "substantive due process." Substance is what process questions are not about.
If the court now "incorporates" the Second Amendment right via the "due process" guarantee, that will be progress because it will enlarge the sphere of protected liberty. And even Justice Antonin Scalia, who recognizes that "substantive due process" is intellectual applesauce, thinks it is too late to repudiate 137 years of the stuff. Still, three points argue for using the "privileges or immunities" scythe against the two gun ordinances.
First, protecting the individual's right to keep and bear arms for self-defense was frequently mentioned by those who drafted and ratified the 14th Amendment, the purpose of which was to protect former slaves and their advocates from being disarmed by state and local governments determined to assault their security and limit their autonomy.
Second, the central tenet of American political philosophy is that government is instituted not to bestow rights but to protect pre-existing rights, aka natural rights -- those essential to the flourishing of our natures. In its 2008 decision, the court affirmed that the Second Amendment did not grant a right to keep and bear arms, it "codified a pre-existing right."
Third, "privileges or immunities" are all those rights that, at the time the 14th Amendment was ratified, were understood to be central to Americans' enjoyment of the blessings of liberty.
Liberals might hope and conservatives might fear that a revivified "privileges or immunities" clause wielded by liberal justices would breed many new "positive rights" -- to welfare, health care, etc. But conservatives know that "substantive due process" already has such a pernicious potential. And they believe that if -- a huge caveat -- it remained tethered to the intent of its 19th-century authors, the "privileges or immunities" clause would be useful protection against the statism of the states.