Q) When Peter saw Jesus walking on water, and thought it was a ghost- does this confirm there are ghosts?
A) There were many superstitions circulating in this time and in this culture. I do not think that Peter believing in ghosts is enough to affirm that ghosts as we expect them to be are a reality. However, we do see an Old Testament story of King Saul consulting a witch to communicate with a spirit of the prophet Samuel. So we see that the conscious spirit of a person can be manifested on earth, but in this case it was administered by an ungodly power. (1st Samuel 28)
We also do not see any evidence in the Bible that the people beyond the grave are paying any attention to what we are doing while on earth. We do see statements made by the dead that express concern for our welfare, and in every example, be it from Luke 16 by somebody in hell, or Revelation 6 by people in heaven, they are always interested in God’s work to be done for us. So if the dead have any interest at all in our earthly experience, it is that we would be pointed toward Christ and the salvation that comes only through Him.
We see from Luke 16 that the dead in hell wish they could communicate to us on earth, but cannot. If they could, they would warn us about the trouble to come if we do not place our faith in Christ.
We see from Revelation that people are aware of the timeline of events on earth, but are most concerned with the coming judgement of the wicked world and the eternal rewards they are excited to receive when we are all resurrected.
I expect if you are in the spiritual heaven, you will be too busy enjoying that to spend anytime haunting people on earth. If you are in hell, you are not able to connect with the people of earth. If people of earth have a spiritual experience they want to attribute to a dead loved one, it is probably the devil or his demons trying to use our sentimentality to distract us from salvation through Christ. Because the dead would have just one message to pass along to us still on earth, and that is to place your faith in Jesus while you have the chance.