Because they never were remotely in the same place. Gollum followed the Frodo out of Moria, down the Anduin until the Emyn Muil, and from then on only was east of the Anduin River. Éowyn spent all of the LotR book in Rohan and in Gondor west of the Anduin River and south of the Emyn Muil.
Who said he did not? Gollum was imprisoned in Mirkwood.
Gollum
Eowyn killed the leader of the Nazgul known as the Witch-king of Angmar .
Eowyn's name is Eowyn. She is the daughter of Eomund and Theodwyn, Theoden's sister. I suppose she could be called 'Eowyn, Eomund's daughter.' She calls herself Dernhelm when she is disguised as a man. Eowyn is played in the films by Miranda Otto.
Gollum. The name was given to him by his relatives, due to the "gollum" noise he made in his throat.
A-O-WIN
Windfola
Their stories simply did not cross, that's all. Treebeard lived in Fangorn Forest, and Gollum lived mainly in the Misty Mountains, and neither of the two ventured out of their homes often. It would have been interesting if he did, though! =) ~Lotrfreak000
Gollum
Bilbo asks Gollum "What's in my pocket" (the Ring) and Gollum cannot answer.
No, Gollum is a victim of evil.
Gollum browser was created in 2005.