I am assuming that the "government" that you are referring to is the same government that administers the social security system, inspects meat prior to sale, plans and builds the roads that we drive on, and provides air traffic controllers so that planes don't crash into one another in the sky overhead. If that is the case, the "government" does not have the power to kill, as you so incorrectly put it.
When a person commits a heinous crime, they are arrested and charged with the crime. The District Attorney, here representing the "people" prosecute the accused with the Defense Attorney representing the best interest of the accused. Originally, the system was weighted (and more so now) so that the people were at a disadvantage in having to prove "beyond a reasonable doubt" that the accused in fact committed the crime. In the case of death penalty cases, their must also be proven facts in the case in which the crime was specifically "heinous".
Here's where you may be getting confused. Twelve people, agreed to by "the people" and the defense listen to the evidence and then decide if the accused is in fact guilty. If they are, then they also must weigh the evidence to ensure that the requirements for heinous acts fit the crime. If they do, then the jury may recommend the death sentence. Depending on what state, the judge may be required to sentence as the jury found, or they may disregard the sentence and impose their own.
Now the "government" gets involved and takes the accused into custody where they are imprisoned for probably about 15 years. The "government", if all of the appeals fail, then administers the sentence imposed by the jury.
By the way, unless you have been in a coma for the last three weeks, I am almost positive that any half-sane person could look at whats going on with our out-of-control government and conclude that it is slightly less efficient that several dead batteries. Big government is a pathway to tyranny and in the IRS scandal, we are seeing exactly that.