In the poem Legal Alien the narrator explains that her dual ethnicity, being both Mexican and a U.S. born American, renders her as a half-citizen in the eyes of those who belong to each of her heritages. 

To the Americans, she will always be considered a Mexican and a Hispanic minority whose first language is not English. Whether she speaks English well or not does not matter because, after all, it is still merely her second language. 

She also experiences detachment from the Mexicans because, since she speaks English fluently and communicates well, they assume that she does not appreciate their daily struggles with the language as "real" Mexican people do. Therefore, she is not Mexican enough in comparison. 

You may speak 
Spanish (but you're not like me") 
an American to Mexicans 
a Mexican to Americans 
a handy token 
sliding back and forth 
between the fringes of both worlds

These verses are clear. Pat is an alien in both groups because neither group can conceptualize that one person can lead a fruitful and successful life as a product of two different cultures. It is possible to assimilate another culture and still revert to our original one. It is possible to be Mexican AND American, without the "hyphen" in between. This is clear to the narrator, but not to those with whom she interacts everyday. 

As far as the narrator's views of her own abilities as a bilingual person, we could agree that she is quite proud of herself. Someone who is able to command two language systems, and shift back and forth in two different societies effectively, should feel quite accomplished. This is evident in the lines:

Bi-lingual, Bi-cultural, 
able to slip from "How's life?" 
to "Me'stan volviendo loca," 
able to sit in a paneled office 
drafting memos in smooth English, 
able to order in fluent Spanish 
at a Mexican restaurant

Notice the choice of words. She is aware of the benefit of bilingualism. She is also aware of the advantages of being a bilingual woman. Those around her may not "get it", but she is quite comfortable in her own skin because she is sure of who she is, and what she can give to society.