Difference between double-sided and single-sided AWGN noise after bandpass filtering?

Consider a bandpass signal $$s(t)$$ with bandwidth $$W$$.

After bandpass filtering, let the output signal be $$r(t)=s(t)+n(t)$$

I have read a paper that denotes $$n(t)$$ as Gaussian noise with one-sided power spectral density $$N_0$$. Therefore, the noise power is $$\sigma^2=\mathbb{E}\{n^2(t)\}=N_0W$$.

What would be the purpose of denoting the noise as single-sided?

It seems that if we consider the noise as double-sided with power spectral density $$\frac{N_0}{2}$$, noise power is still $$\sigma^2=\mathbb{E}\{n^2(t)\}=N_0W$$ since we have to integrate over the negative frequencies and the positive frequencies.

What is the purpose of describing AWGN noise as single-sided versus double-sided? Considering the case of real signals, do both end up giving the same results?

Both descriptions give the same result. We often use the one-sided noise power spectral density (PSD) because for real-valued processes the negative frequencies are redundant, so defining the PSD for positive frequencies is sufficient. You just have to scale the noise power spectrum such that integrating the one-sided power spectral density (PSD) over the positive frequencies gives the same result as integrating the two-sided PSD over positive and negative frequencies. I.e., for white noise with a constant PSD, defining the one-sided PSD as $$N_0$$ means that we would need to define the two-sided PSD as $$N_0/2$$ such that the noise power remains the same: $$N_0W=N_0/2\cdot 2W$$