If someone grows up bilingually (say, a Mexican-American who's a native speaker of both English and Spanish), and they learn a third language (say, Mandarin), what accent will they speak it with? (Or maybe it varies depending on the speaker.)
As the other poster indicated, accent is the application of native phonology to another language.
However, if someone grows up speaking two languages, they would necessarily have less of an accent than someone who speaks just one language. Two languages necessarily have a larger phoneme inventory than just one, which means the speaker has more native phonemes to choose from.
The perfect example is from Portuguese. In my experience monolingual Spanish speakers have a really hard time pronouncing European Portuguese sounds, particularly the reduced vowels and opened variations of E and O.
Whereas English speakers have trouble pronouncing rolled R and reduced consonants such as the approximates of B, V and D.
In my experience, students who grew up speaking Spanish and English do wonderfully learning the Lisbon variant of Portuguese.
In the case of Mandarin, from what I've heard the Chinese usually classify foreigners differently. Since that language is so unrelated to Spanish and English, typically they just say someone had a Western accent, which it is usually not easy to determine the origin.
Having an accent while speaking a foreign language is just applying the phonetics of the native language to the new one.
Having said that, I would assume that in a situation when a bilingual speaker is equally fluent in both his native languages, the phonetic system which is going to be applied depends on the language he uses more often.
But that is just my personal guess, feel free to criticise it with some references.