The Fifth Amendment, and all the other amendments in the "Bill of Rights" (numbers 1-10) were universally understood when passed to be restrictions on the Federal Government only. The courts treated them that way through the end of the US Civil War. This was made definite in the US Supreme Court case Barron v. Baltimore, 32 U.S. (7 Pet.) 243 (1833)
Since the passage of the Fourteenth Amendment the courts have decided that most of the provisions of the bill of rights also apply to actions by the states. A few do not apply, such as the requirement that indictments be by a grand jury, and the Third Amendment ban on quartering soldiers in private dwellings.
This was done through a somewhat roundabout mechanism -- the Supreme Court decided that the protections of the Bill of Rights were included in the Due process clause of the 14th. As a result not all the provisions were made applicable at the same time. (Most were held to be incorporated during the period from 1925-1985. Gitlow v. New York, 268 U.S. 652 (1925) thru Roberts v. United States Jaycees, 468 U.S. 609 (1984)).
Modern legal opinions sometimes discuss incorporation as if it was a fact from the passage of the 14th in 1868. But the actual gradual process is clear in the case law. For example, the Sixth Amendment right to counsel in criminal cases was first incorporated in Powell v. Alabama, 287 U.S. 45 (1932), but only for death penalty cases, and only if "special circumstances" existed, such as a defendant who was illiterate, far from home and support, or feeble-minded. Later cases gradually found "special circumstances" in more and more fact patterns, and in Gideon v. Wainwright, 372 U.S. 335 (1963) the Court extended the right to all felony cases. It has later been extended to misdemeanor cases if jail time is a possible result. A similar history could be spelled out for the Fifth Amendment's protection against self-incrimination, or for the Fourth's against search and seizure, particularly the "exclusionary rule".
I, and a number of legal scholars who have better rights to an opinion, think that the 14th's "Privileges and Immunities" clause would have been a more sensible means to this end, but for various reasons that isn't how it was done. Justice Thomas seems to be trying to reverse this -- he has made comments in a number of opinions of late that various things should be protected under the Privileges and Immunities clause of the 14th, rather than the Due Process clause.. Even if the Court adopts this theory, it probably won't change many outcomes. That is how the Fifth, and other Bill of rights Provisions like the Fourth (search and seizure) and the First (free speech and religion) have been applied to restrict the states.
None of these provisions directly restrict private individuals. In some cases, courts have said that while individuals may not be forbidden to do things that are forbidden to governments under the Bill of Rights, the courts will not help you do such things, such as by enforcing contracts to do them.
No person shall be held to answer for a... crime
"held to answer" here means prosecuted in court. Only governments do that. That provision forbids criminal court cases that do not start with a grand Jury indictment. it is one of the few Bill of Rights provisions which the Supreme Court has held do not apply to the states. But in any case it is purely procedural. It doesn't say that crimes may not be prosecuted, nor that they must. It says only "if you want to try someone for a crime, this is a step you must go through." The other provisions of the Fifth all do apply to the states, such as the ban on double jeopardy, and the protection against self-incrimination.
does that mean that government can declare it legal for citizens to kill a particular person?
No. That would violate the Fifth Amendment's Due Process clause if don3 by the Federal Government, and the Fourteenth Amendment's Due Process clause and its Equal Protection clause if done by a state. It would probably also violate the provision against Bills of Attainder, and perhaps the provision against cruel and unusual punishment.
Once upon a time, several hundred years before the US was founded, the government of England did just that. It was called "outlawry". For certain crimes, the punishment was to be put "outside the law". An "outlaw" (in this older sense) was not protected by the law. Anyone could kill an outlaw, or steal from one, and the legal system would do nothing about it. The US has never used outlawry.