In most cases, the currency is converted by the receiving bank. In such a scenario, the sending/receiving bank have a direct correspondent agreement and the receiving bank provides the exchange rate into the local currency and does the conversion.
In the event a middle, correspondent bank is used, same result again, the receiving bank provides the forex rate and does the conversion.
In some cases, there are aggregators involved, in which case, depending on the arrangement, they might already covert for you and then do an onward settlement with the receiving bank in local currency.
In some cases, the sending bank provides the rate (based on the liquidity of the corridor), though more and more this is less practiced on exotic currency pairs.