I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal distribution $N_2$ with mean $m_2$ and standard deviation $\sigma_2$. In this case, what will be the combined or joint probability distribution of $X$? Will it be $N_1+N_2$ or $N_1 N_2$ ?
(Addition) Let's assume $A$ and $B$ are independent measurements. We can think about a situation when a measurer $A$ comes and measure the distribution of $X$ and then next a person $B$ comes and measures the distribution again. Both measurers measure independently. The question is what will be the true probability distribution of $X$ in this case? We assume that the measurement of $A$ and $B$ are equally reliable.
(Paraphrasing a comment from r.e.s.) If person C receives reports from equally-reliable observers A and B, stating their respective independent judgements about X (in the form of the stated normal distributions), then how does C combine these reports to form a fair and unbiased judgement about X?