Is it permissible to use dishes washed by non-Muslims in a shared kitchen in a non-Islamic country, or is it obligatory to wash them three times for purification?

It is permissible to use the utensils of non-Muslims, as the default state for them is purity. The Prophet, peace and blessings be upon him, and the Companions used them. If it is known that they cook pork or carrion in them, or drink wine from them, it is preferable to avoid them unless nothing else is available. In that case, they should be washed, and one may eat and drink from them. It is not required to wash them three times; rather, it is sufficient for the trace of impurity to be removed. It is also not a condition that a Muslim performs the washing.