In the New Testament, there are several meanings attached to 'power'. It can mean ability - dunamis. Or, to give power - didomi. Or, strength - kratos. Or, strength, force - ischus. Or privilege, authority - exousia. So, what word is used in the text in question? It is dunamis - ability, power.
The text (as quoted in the OPs comments) is perfectly clear. Those waiting disciples were to be given the gift of the Holy Spirit from heaven, which would empower them for a specific purpose, as the rest of Jesus' sentence details:
"...and you will be my witnesses in Jerusalem, and in all Judea and
Samaria, and to the ends of the earth."
It was specific power for a specific purpose - to be given the ability to carry out the great commission that will remain in force for all Christians until Christ returns in glory. Clearly, the gifted power would not be restricted to just those few disciples who witnessed the resurrected Christ ascending back to heaven just after he spoke those words of command and assurance. The task of reaching out into all the world to witness to the risen Christ has been on-going for near-on two thousand years so far.
This is backed up by what the New Testament says about this particular power of God in Romans 1:16. The writer was the apostle Paul, who had not been present for the Acts 1:8 declaration, but who most certainly did become a witness of the risen Christ, being given power/ability of the Holy Spirit to testify far and near as the great commission required. He wrote:
"That is why I am so eager to preach the gospel also to you who are at
Rome. I am not ashamed of the gospel of Christ, for it is the
power [dunamis - ability] of God for the salvation of everyone who believes; first for
the Jew, then for the Gentile." (emphasis mine)
There have been, and still are, many people who claim to be Christians, but who do not concern themselves to testify (witness) to the good news (gospel) of a crucified and resurrected Christ, who is Saviour and Lord to all who have faith in his finished work. It may be stating the obvious, but those who do not carry out the great commission just do not have the power of the Holy Spirit, given for that task. Some almost appear to be ashamed of the gospel of Christ, not wishing to be publicly identified with it, contrary to Romans 1:16.
Of course, there are myriad ways in which testifying to the gospel of Christ can be done, both by those who stand up in public to proclaim an unadulterated, biblical gospel, and via other Christians who quietly testify to the power of the risen Lord by words and deeds in the everyday course of their ordinary lives. "The Lord knows those who are his, and everyone who confesses the name of the Lord must turn away from wickedness." - 2 Timothy 2:19
We can observe claims and see the way lives are lived, and whether words spoken stick to the pure, biblical gospel of Christ. However, the promise is sure, that where the power of the Holy Spirit enables faithful preaching and witness, believing souls will be saved. The responsibility of individual Christians is to ensure we are faithful to the great commission, the Holy Spirit enabling us - giving us the ability - to testify, the Spirit producing miraculous results, as Christ is lifted up and God praised.