Assalamu Alaikum. 1) I would like to know if it is permissible for anyone to quote the Prophet, sallallaahu 'alayhi wa sallam, as saying so and so without providing references (even if it is something good)? 2) Is it against Islamic etiquette to request references from the informer who has not provided references after saying that the Prophet, sallallaahu 'alayhi wa sallam, said so and so?
Could you answer these for me (with references) as I am also afraid of giving incorrect information to others? I would greatly appreciate it. Jazak Allah Khayra.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

Muslims should bear in mind that attributing statements and reports to the Prophet is not the same as attributing them to any other person. They should cautiously and keenly verify the authenticity of any words before attributing them to the Prophet . He said, “Whoever (intentionally) ascribes to me what I have not said, then (surely) let him occupy his seat in Hellfire.” [Al-Bukhari]

It should be noted that if one wants to cite a Hadeeth as evidence, he is not required to mention the source; however, if one is unqualified to judge whether a Hadeeth is authentic or weak, he should not cite any Hadeeth as evidence except after verifying its authenticity and soundness for evidence by consulting scholars or reading that in their standard reference books and resources.

There is no harm in asking for references from the person quoting a Hadeeth as evidence; however, it is impolite to do so with a reliable scholar who is known for accuracy and keenness on verifying the authenticity of Ahaadeeth before quoting them. In fact, it is impolite to ask such a scholar to provide evidence at all.

An-Nawawi said, “The ordinary Muslim should not ask the Mufti for evidence, and he should not ask, "Why did you say … ?" If he wishes to know the evidence to be reassured, he may ask the Mufti in another meeting or in the same meeting but after accepting the Fatwa in its bare form.” [Aadaab Al-Fatwa wal-Mufti wal-Mustafti]

Allah Knows best.