After decomposing my data, should I fit an ARIMA to the remainder or the trend?

I've decomposed my data to get rid of the seasonality. Now I want to use the arima function to fit an ARMA(p,q) model.

Do I fit the model to the "random" component of the decomposition? That seems odd to me, as surely it's just white noise?

Or should I fit the model to the "trend" component?

You should fit ARIMA to the remainder. Indeed, if your trend is linear, it's completely described by a fomula like $at+b$ where $a$ and $b$ can be estimated by least squares (and this doesn't contain AR/MA parts).
Moreover, ARIMA on the remainder is precisely useful when residuals $\eta_t$ of the linear regression $X_t = at+b+\eta_t$ aren't pure white noise and do contain some non-trivial autocorrelation (which can be modelled by ARIMA).