To compensate for the poor reliability of Physical Unclonable Function (PUF) primitives, some low complexity solutions not requiring error-correcting codes (ECC) have been proposed. One simple method is to discard less reliable bits, which are indicated in the helper data stored inside the PUF. To avoid discarding bits, the Two-metric Helper Data (TMH) method, which particularly applies to oscillation-based PUFs, allows to keep all bits by using different metrics when deriving the PUF response. However, oscillation-based PUFs are sensitive to side-channel analysis (SCA) since the frequencies of the oscillations can be observed by current or electromagnetic measurements. This paper studies the security of PUFs using TMH in order to obtain both reliable and robust PUF responses. We show that PUFs using TMH are sensitive to SCA, but can be greatly improved by using temporal masking and adapted extraction metrics. In case of public helper data, an efficient protection requires the randomization of the measurement order. We study two different solutions, providing interesting insights into trade-offs between security and complexity.