Is it permissible for a needy person who is ashamed to ask for money to request money from his friends claiming it is to help another needy person, while he himself will take the money?

Merely losing money does not permit begging. Begging is permissible for someone who has been afflicted by severe need in their expenses and the expenses of their dependents, or who has many debts they are unable to repay. The Messenger of Allah, peace and blessings be upon him, said: "O Qabisa, begging is not permissible except for one of three: a man who has taken on a debt [on behalf of others], for him begging is permissible until he obtains it, then he stops; a man who has been afflicted by a calamity that has destroyed his wealth, for him begging is permissible until he obtains enough to live on – or he said: enough to meet his needs; and a man who has been struck by poverty, until three people of sound judgment from his people attest, 'So-and-so has been struck by poverty,' for him begging is permissible until he obtains enough to live on – or he said: enough to meet his needs. As for anything else, O Qabisa, it is unlawful gain that its owner consumes unlawfully."

What friends give of money, if they direct it to a specific person or give it to someone who possesses a specific characteristic of need, then it is not permissible to spend it in a way other than what the givers intended. If you are in need to the degree mentioned in the hadith, and your friends do not earmark their donations for specific individuals, and you possess the characteristic they desire, then there is no objection to taking what they give you.