so if I understood you correctly you want to show the user's current location on the map without paying to any third party, if that's the case you need to do the followings.

1. Get user latitude and longitude.
2. Show those latitudes and longitudes on the map using some marker.

1. Get user latitude and longitude.

You can follow the below article for getting user latitude and longitude.

• ReactNative: https://aboutreact.com/react-native-geolocation/
• Expo: https://docs.expo.dev/versions/latest/sdk/location

2. Show those latitudes and longitudes on the map using some marker.

So in this step, we will be using react-native-maps and OpenStreetMap.

Note: Do not change urlTemplate in the UrlTile

Here is the reference for custom map tile from react-native-maps

https://github.com/react-native-maps/react-native-maps#tile-overlay-using-tile-server

import MapView, { UrlTile, Marker } from "react-native-maps";

const { width, height } = props;
const ASPECT_RATIO = width / height;
const LATITUDE_DELTA = 8; //Very high zoom level
const LONGITUDE_DELTA = LATITUDE_DELTA * ASPECT_RATIO;
const userCoords={{
   latitude: number,
   longitude: number
}};
<MapView
  initialRegion={{
    latitude: 51.1657,
    longitude: 10.4515,
    latitudeDelta: LATITUDE_DELTA,
    longitudeDelta: LONGITUDE_DELTA,
  }}
  loadingEnabled
  minZoomLevel={3}
  maxZoomLevel={10}
  showsIndoors={false}
  showsTraffic={false}
  showsBuildings={false}
  showsScale={true}
>
  <UrlTile
    /**
     * The url template of the tile server. The patterns {x} {y} {z} will be replaced at runtime
     * For example, http://c.tile.openstreetmap.org/{z}/{x}/{y}.png
     */
    urlTemplate={"http://c.tile.openstreetmap.org/{z}/{x}/{y}.png"}
    /**
     * The maximum zoom level for this tile overlay. Corresponds to the maximumZ setting in
     * MKTileOverlay. iOS only.
     */
    maximumZ={10}
    /**
     * flipY allows tiles with inverted y coordinates (origin at bottom left of map)
     * to be used. Its default value is false.
     */
    flipY={false}
  />
  <Marker coordinate={userCoords}>
    /**
     * Your custom marker here 
    */
  </Marker>
</MapView>;

You currently can't do that with react-native, you have to add your own map. You can use react-native-maps