What you're describing concerning personal prophecy is called a "word of knowledge" or a "word of wisdom." The Apostle Paul talks about these gifts in 1 Corinthians 12. We also see these gifts operational in the book of Acts.
Such prophetic utterances are very biblical but they are also greatly abused in the Body of Christ today. Personally, I am very wary of receiving a prophetic word from anyone I don't know or anyone who doesn't have a proven track record of maturity and prophetic sensitivity.
Additionally, we are exhorted to "test everything" and that especially includes prophetic messages. There are many ways we can test such things, such as:
1. Through the inner witness of the Spirit. Does the message resonate in my heart as true, or does it give me a check in my spirit?
2. Through the Word of God. Is there anything in the message that violates what I know to be true in the Bible?
3. Through the test of time. If the person spoke to me of a coming event, did it come to pass as they said?
4. Through prayer. We should always pray about any personal message that is supposed to have come from the Lord and I would encourage you to ask the Lord to confirm it through other sources. If God is speaking to you He doesn't mind repeating Himself for the sake of clarity and confirmation.
Anyone who becomes annoyed or angry when we tell them we will test their message to see if it holds up is not operating in the Holy Spirit and should be avoided. Not everyone who claims to have a message from the Lord is truly walking in the Spirit.
Prophetic messages are a wonderful way for the Lord to encourage or warn us, but we must be diligent to test everything.