General American, also known as Network English is what you could deem as "neutral" English. It is the English used by news reporters and radio personalities to maintin the objectivity and the impartiality in news reporting. In real life, however, there is no neutral English and the ACTFL does not recognize any English accent as considered "proper.
A native speaker of English is a person who has been raised speaking the English language, regardless of the place where he or she was born. It is identified as a a person who learned English as their first language, and continued speaking it throughout their lives.
Local English is the accent that is acquired by the group which adopts it. It has everything to do with the unique traits shared by a group of speakers, and requires more than mere repetition: It is a fragment of the culture and temperament of group as well as what gives the group its identity.
Neutral English is, I believe, for the most part an artificial English in which regional differences in pronunciation and vocabulary have been levelled out to create a language that doesn't pin the speaker to any one particular region or social class. TV newspeople in the United States and elsewhere (although less frequently now, perhaps) learn to downplay their regional versions of English while speaking to the camera. Many of the university students that I encountered while teaching in Germany who spoke English well also spoke a neutral English.
Native speaker English, by contrast, is the real, living language as it is manifested in the daily communications of people who grew up speaking the language. With native fluency comes a seeming inborn (it's learned, of course!) sense of what sounds right and what doesn't, such as where to place adverbial phrases in a long sentence.
Local English, as best as I understand, is any pattern of English that is used in a way that marks the person's origins and/or identity, whether in terms of geography, social class, ethnicity, and any number of other social positions. Some groups use words that other groups don't, and the same word can be pronounced differently in different groups. In my local English, for example, I don't use the word "po-po" (which is a common enough word for many speakers of black vernacular Engish) and I stress the second syllable of "police" when used as a noun (pu-LIS) when many speakers of black vernacular English would stress the first syllable (PO-lis). Local English, as I understand it, isn't necessarily Native Speaker English; for example, I would also characterize (in admittedly very broad terms) the English that is spoken by non-native speakers in India as local English.
One goal of sociolinguistics is to move past the idea that some people speak a correct version of the language and others do not. Instead, this approach looks at which versions of a language are spoken by which groups and how those versions are differently valued. Some versions of English (again, local English) have much more status that others. The link below shows that RP (a modified middle-class version of English spoken in England that has been cultivated for well over a century) continues to have more status in the eyes of many non-native speakers than does neutral American English.
Neutral English - The Art of Speaking & Writing without Regionalism
Neutral English training includes the following modules.
2. Jaw Exercises
3. Vowel & Consonants Sounds
4. Syllable and Word Stress
5. Sentence Formation
6. Working on Thought process
7. Voice modulation & Speech therapy
8. Articulation Exercises
1. Write in international style and standard.
2. Stay away from jargon your reader may not understand.
3. Pay attention to grammar.
4. Refer World famous English literature, books & Novels
5. Punctuation, selection of words, narration and context are very important.
Neutral English: This is not to imply that neutral pronunciation has greater merit than any of the regional dialects. It is also a dialect, but one without any regionalisms. It is, however, the dialect that is used by trained speakers and performers for public usage. The way we speak English regionally is part of our personal identity. It is something that should be used and mentioned in our everyday speech. However, when we are speaking or performing in a public forum, neutral English should be used, so as to erase regional barriers and communicate effectively with the most people. - neutralenglish.com
Local English is normally used with slangs and dialect, lacks respect. Native speakers English is also local English in a broad sense, respectful.