On what does it depend? Different agencies have different policies. It also depends on where someone serves. Some host countries don’t care how missionaries receive their income, as long as their income doesn’t take away jobs from the local economy. Other countries welcome missionaries as missionaries only if their financial support comes from an outside source.
Another factor ties in with Jesus’s comment about serving two masters (Mt 6:24, Lk 16:13). I’ve observed that when people self-fund through whatever means, whatever the source of income becomes the primary focus, and the ministry they’re there to do gets put on the back burner. Cross-cultural ministry is HARD, and it’s harder yet when attention is so divided.
It is entirely possible to rotate work and ministry, depending on location and org., but there are downsides to that model. One is that the missionary never fully engages the host culture (why should you die to self learning the language and culture if you’re only there a short while?) and the host culture people never fully receive the missionary (why should they if they know we’re leaving soon?). Communal cultures are typically warm, but slow to engage; they value time differently than we do and they see Western efficiency as cold hearted. The continuity of being present for at least two unbroken years allows for relationship building, and for reacquaintance upon return.
Raising funds is no easy thing, but God does an incredible work in the missionary by training the daily trust in His provision and that work of God is observed by the people. How can we invite people to trust God daily when we ourselves do not?
Rotating between home country and int’l service: There is a great precedent for this as mission hospitals continually need healthcare workers who can fill in for a season, either so missionaries can get a much-needed vacation or are out due to health reasons or they need to go to a training. Another consideration would be to serve during certain seasons that have heavier patient loads, like malaria season. There’s also a precedent for healthcare workers to come for a season to teach, train, and/or supervise national healthcare workers or residents for a season as well. Healthcare workers are also needed for short-mid length terms as missionaries go on home assignments which typically last anywhere from 3 months to 2 years. *Most mission agencies still require mission workers to raise enough funds or support to cover travel, health and medivac insurance, and it’s wise to also raise “ministry funds” to help do special projects in the community as well. Raising prayer support is equally important and should be required by each agency. Those taking this approach should plan on receiving some form of cross-cultural training as well.