A user asked this question and then withdrew their post. Re-posting the answer here because I thought it was a valid question that others may have in the future:
Near me can be implemented with a Geo Filter: Filters | Swiftype Documentation.
"filters": {
"location": {
"center": "37.386483, -122.083842",
"distance": 300,
"unit": "km"
}
}
Pass the user's location in as "center" and then adjust the "distance" and "unit" to whatever you would consider "near".