Among believers who say that guardian angels may choose for themselves how to best answer people's prayers are: Hindus, Buddhists, astrologists, and those who practice New Age spirituality.
Among believers who say that guardian angels only carry out God's will when answering prayers are: Jews, Christians, Muslims, and Sikhs.
In Hinduism, angelic beings called devas carefully consider prayers that Hindus either say aloud or write down and leave near statues of the devas in temples and shrines. Hindus believe that devas sit together in groups called sangam circles to transfer their thoughts about people's prayers among their minds and reach a consensus about how best to answer each prayer in light of the karma of the person who prayed it. Then, once all the devas agree about how to answer the prayer, one of them leaves the sangam circle and travels to Earth to help answer the prayer.
In Buddhism, angelic beings called bodhisattvas help answer prayers by adding their good thoughts to the prayerful thoughts that Buddhists express through meditation, thereby strengthening the power of those thoughts to achieve good results.
In astrology, each person has three guardian angels who have special power to answer that person's prayers: the Incarnation Angel, the Heart Angel and the Intellect Angel. The particular angels who serve in those roles for the person are determined by the the date and time of that person's birth.
In New Age spirituality, guardian angels regularly answer people's prayers either by manifesting the results, or by guiding the people who pray to help them find the answer for themselves.
In Judaism and Christianity, guardian angels carry out God's will when answering prayers. However, angels may get involved in helping people plead their cases before God to try to influence his will and how it's carried out. Genesis chapter 32 of the Torah and Bible records how the prophet Jacob wrestled with an angel overnight to try to convince God to give him a blessing, and the angel finally tells him that God has decided to give Jacob the blessing he had been seeking. Verses 26 through 28 describe how the struggle ends: "Then the man [angel] said, 'Let me go, for it is daybreak.' But Jacob replied, 'I will not let you go unless you bless me.' The man asked him, 'What is your name?'. 'Jacob,' he answered. Then the man said, 'Your name will no longer be Jacob, but Israel, because you have struggled with God and with humans and have overcome.' In Christianity, Jesus Christ encourages people to develop a close relationship with him so that God's desires will naturally become their own and they'll want to pray for God's will and see it come true in their lives: "If you remain in me and my words remain in you, ask whatever you wish, and it will be done for you," he says in John 15:7 of the Bible.
In Islam, guardian angels answer people's prayers only according to God's will. The Qur'an declares in chapter 66 (Al Tahrim) verse 6 that angels: "disobey not the commands they receive from God, but do that which they are commanded."
In Sikhism, as well, guardian angels carry out God's will only when answering people's prayers. The Japji, a song about God in prayer form that was composed by Guru Nanak Dev, the founder of Sikhism, says in hymn 30 about God: "Whatever He wishes, He makes to happen according to His Command."