All of the following would require a programming effort (as far as I'm aware).
Quote:could be possible with PineTime to send real time heart rate sensor data [..]
The PineTime has functionality that allows it to wake periodically (see the new function to show the time each hour) and execute some function. It should be possible to wake the I2C bus, wake the HR sensor, make a reading and go back to sleep - all without entirely powering the device back on.
Quote:[..] through bluetooth connection directly (without phone) to a local server ?
Kind of. You need to get your data from the watch to the server via Bluetooth, so this could happen in bulk either via a Phone or Desktop. The good news is though that it should now be possible to collect HR data without needing to be tethered all the time to another device.
Note though that some important data such as GPS will not be available without a complimentary device offering GPS data.
In the future it could be great to see a PineTimePro that has a little more hardware capability, making it more independent from the need for external devices. Support for WiFi, a magnetometer, GPS and LoRa are all in theory within the realm of possibility - but expect a much more expensive device to result.