How big must an earthquake be in order to be detected on the other side of the Earth?
No matter how large the earthquake is, no human could ever "feel" an earthquake on the opposite side of the earth. For example, after the 2004 Sumatra earthquake (M 9+), the ground was vertically displaced a minimum of 1cm everywhere on Earth. Such a displacement should have been perceptible to humans. However, this occurred over hundreds of seconds and people just aren't sensitive to such long period displacements (seismograms showing true ground displacement at scale are attached). For a nearby event, a "felt" earthquake is generally considered to be ~M 3, slightly less if the conditions are just right.
However for intruments the answer depends somewhat on where you are and what you are "listening" with. An AS-1 (educational seismograph) in a classroom will have a different experience than a good broadband station like those used in the Global Seismographic Network (GSN). Further, a good GSN station at a quiet site on the continent will also be different from a GSN station with identical equipment, but located on a noisy island e.g Easter or Pitcarin Island (noise can be both man made, trains etc, or natural, wave action, wind etc).
For a quiet GSN stations, you can see magnitudes in the mid-4s under good circumstances, low 4s if exceptional. For noisy sites, it might be in the mid-6s or higher. This sensitivity can be enhanced even further if you have a seismic array (multiple seismometers deployed at the same time). When using a seismic array, seismologist digitally "stack" the data from multiple stations to improve the signal to noise ratio and can routinely detect events in the mid-4s, and reach 4.0 under good circumstances.
Going a bit beyond instrumentation and noise, there are other factors that can allow you to detect small events globally. There are "sweet" spots on the "other side" including the PKP caustic near 140° and at the antipode (the opposite side of Earth from the epicenter). At the antipode, there are focusing effects that amplify all the energy traveling around and through the earth at that one spot.
Some clarifying information:
PKP - This nomenclature refers to a P wave that travels through the Mantle (P), then through the outer core (K), and then back out through the mantle (P) = PKP. The PKP caustic is a focusing effect of P waves that left the source at uniform angles but become concentrated as they refract differently in the outer core depending on the depth they penetrate the outer core (see the attached figure showing ray paths traveling through Earth for the PKP caustic)
Antipode - Here are some data visualizations of ground motion as recorded by the USArray. To better understand the visualizations, please view the scaffolded tutorial
- Earthquake epicenter in the center of the array
- Antipode just outside the edge of the array