Your question is posing a false dichotomy. Many legal theorists would argue that strong contract and tort law that supports individual rights is thereby supporting the public interest.

In general, American contract law provides a general framework for private parties to enter into and enforce agreements. Tort law protects individuals by providing a remedy for personal harms. In both cases, serving the needs of the individual citizen also serves the needs of the public at large.

Contract and Tort law are from the common law. The main purpose of contract law is to encourage people to make binding agreements, it usually dealt with employer-employee relationship types.

Enforcing agreements legally creates a more stable society overall. It keeps people from "crossing their fingers" to their own benefit and allows for people receive damages or force the person to fulfill the promise they made.

Tort laws were the same. It allowed for people to receive compensation for wrongs done to them. In this sense the two areas of law protect the public interest.

Tort law is based on rights of property and freedom from physical injury. So they definitely protect individual rights.

Contracts can cause people to give up certain rights, but its based upon a voluntary agreement. One can gain rights because of a contract and also lose rights, or both at the same time. It depends on the agreement, but sacrificing rights or gaining them in an agreement might not be as good or bad as you think. One might have to give up the right to have a managing interest in a partnership, but they don't have to invest as much money and possibly be less liable for monetary losses. One may gain the right to leave a contract at will whenever they want but that may mean sacrificing something else.

IANAL, so the more general definitions of contract and tort law are best left to someone else, but I will say that in antebellum era Supreme Court cases, the nature of contracts was at times pitted against the (perceived) public good, and how it was handled depended on who was Chief Justice at the time.

Two main examples are Dartmouth College v. Woodward, presided over by John Marshall, and Charles River Bridge v. Warren Bridge, presided over by Roger Taney before he made the Dred Scott decision.

In Dartmouth, Marshall upheld the absolute inviolability of contracts over the arguments that it needed to be declared void for the public good. One could argue, as Marshall certainly did, that the sanctity of contracts was for the public good, but one can see the nuance at play here.

In Charles River Bridge, Taney ruled (in his mind) in favor of the public interest and against the inviolability of contracts.