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According to this the following copies weights from one model to another:

target_model.set_weights(model.get_weights())

What about copying the weights of a specific layer, would this work?

model_1.layers[0].set_weights(source_model.layers[0].get_weights())
model_2.layers[0].set_weights(source_model.layers[0].get_weights())

If I train model_1 and model_2 will they have separate weights? The documentation doesn't state whether if this get_weights makes a deep copy or not. If this doesn't work, how can this be achieved?

Of course, it would be a copy of the weights. It does not make sense the weights object to be shared between two separate models. You can check it for yourself with a simple example like this:

model1 = Sequential()
model1.add(Dense(10, input_dim=2))

model2 = Sequential()
model2.add(Dense(10, input_dim=2))

model1.compile(loss='mse', optimizer='adam')
model2.compile(loss='mse', optimizer='adam')

Test:

>>> model1.layers[0].get_weights()
[array([[-0.42853734, 0.18648076, -0.47137827, 0.1792168 , 0.0373047 ,
 0.2765705 , 0.38383502, 0.09664273, -0.4971757 , 0.41548246],
 [ 0.0403192 , -0.01309097, 0.6656211 , -0.0536288 , 0.58677703,
 0.21625364, 0.26447064, -0.42619988, 0.17218047, -0.39748642]],
 dtype=float32),
 array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], dtype=float32)]

>>> model2.layers[0].get_weights()
[array([[-0.30062824, -0.3740575 , -0.3502644 , 0.28050178, -0.68631136,
 0.1596322 , 0.08288956, -0.20988202, 0.34323698, 0.2893324 ],
 [-0.29182747, -0.2754455 , -0.64082885, 0.29160154, 0.04342002,
 -0.4996035 , 0.6608283 , 0.10293472, 0.11375248, -0.43438092]],
 dtype=float32),
 array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], dtype=float32)]

>>> model2.layers[0].set_weights(model1.layers[0].get_weights())
>>> model2.layers[0].get_weights()
[array([[-0.42853734, 0.18648076, -0.47137827, 0.1792168 , 0.0373047 ,
 0.2765705 , 0.38383502, 0.09664273, -0.4971757 , 0.41548246],
 [ 0.0403192 , -0.01309097, 0.6656211 , -0.0536288 , 0.58677703,
 0.21625364, 0.26447064, -0.42619988, 0.17218047, -0.39748642]],
 dtype=float32),
 array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], dtype=float32)]

>>> id(model1.layers[0].get_weights()[0])
140494823634144

>>> id(model2.layers[0].get_weights()[0])
140494823635664

The ids of kernel weights arrays are different so they are different objects, but with the same value.