Disclaimer: not a historian, and not a Southerner, so pardon my spellings and feel free to correct me.

This is a very good question. A small piece in a snapshot of history that requires a firm hold in tremendous amount of background to explain. I would try to keep this accessible but please bear with me, I have to give primers to some complex topics for this one to cover the bases.

The very first thing to consider is the fact that back in the 19th century, there was no real fiat money: prior to the establishment of the Bretton Woods system well into the 20th century, the world functioned with precious metals as their backbone. Silver had been historically used in many parts of the world since the Spanish Empire began exploiting the gigantic reserves in the Americas; Spanish dollar, and later Mexican dollar, thus became the first global currencies. In fact, the American Dollar originated as legally defined equivalencies of the Spanish dollar since its inception and the Spanish dollar remained as legal tender until 1857. Although gold began to catch up, it never dethroned silver and most countries in the world would keep both reserves and functionally accept both as currencies. Particularly, both the USA and China were using silver standard to different degrees. (Even though America switched to gold standard in the late 19th century, public confidence still rested on silver - hence the USD notes were "silver certificates").

From a sociological perspective, prior to the establishment of modern, nationally integrated markets, each local market would establish their own standards, norms and traditions. In Qing China without such nationalized financial systems, a huge variety of currencies were used that may differ from place to place, but the universally accepted currency are the metals copper and silver - with silver "Liang" (Tael) is weight normalised for large purchases, silver coins (dollars - Mexican, Spanish or American) as intermediaries, and copper coins for day-to-day purchases. Only silver was precious metal and held value reliably. This also allowed for a booming domestic banking industry: since silver holds value, it could be recognised and exchanged across provinces by private bankers keeping their own books, who could also exchange American silver dollars to copper coins or other currencies. These banks were not regulated and operated on a basis of credit and trust. So by now you would realize that, yes, you can indeed spend American money in China.

Now we covered the elephant in the room, it is time to talk about how Chinese labourers emigrated abroad, which differed on their geographic origins, social classes and the time of arrival. Since the Ming Dynasty, there had already been presence of Chinese diasporas in Southeast Asia, namely farmers and merchants from Guangdong and Fujian provinces in (later) British Malaya and Dutch East Indies, who already established institutions named "侨批" or Remittances Bureaus, which functioned on a trust basis that connected Fukien and Cantonese clans to their overseas branches, and was the first wave of Chinese emigration. Although at the time, monetary remittances was not the top concern. But as China entered a period of conflict and chaos in the middle of 19th century, the landscapes in the South drastically changed and many found themselves in dire need of new ways to survive. This marked the beginning of the second wave of Chinese diaspora emigration, and it was global.

The first type of labourer were kidnapped slaves; they were typically taken by bandits or pirates and sold for slavery across the world, primarily plantation colonies like Cuba. They were virtually non-existent in the United States. The second type of labourer were indentured, long-term contract workers; they would sign terms that limit their mobility and personal freedoms to plantation owners, sometimes for life. They came as some of the earliest labourers to the US under the historical background that their hometowns in the Southern provinces were suffering from tremendous combinations of natural disasters and the Taiping Rebellion, and they quite literally did not have another choice. They were mostly able to escape the fate due to these clauses being unenforceable in the free state of California (further upheld federally by the Anti-Coolie Act of 1862) and pursuit their own employments, with the exception of Hawaii where the geographic isolation and lack of legal protection made many Chinese labourers effectively enslaved for their lifetimes. The intention of these indentured labourers was to drop off extra mouths for their clans during the time of crisis, and they were not intended to be paid extra. But it was these pioneers who were the first to land in the United States; they also became some of the first to open up immigrant enterprises.

But a decade later, the economic situation of the costal provinces began to improve, coupled with the discovery of gold in California (fun fact: San Francisco translates to "Old Gold Mountain" in Chinese, one of the rare exceptions to the Local Pronunciation Doctrine because it predated the doctrine, as it was one of the earliest Chinese communities outside Asia. Extra fun fact: the New Gold Mountain is Melbourne). Many Chinese men in the coastal provinces began to proactively seek out opportunities in the US to earn high wages, and this is the period I reckon OP is referencing to. In the meantime, construction of infrastructures in the US West Coast such as railways induced a massive labour shortage, and many American businessmen and some government representatives began to actively recruit Chinese labourers. But the cost to emigrate to the US was about 70 dollars, more than they could ever earn in their lifetimes. So, some American business owners used their existing business connections in China, and teamed up with the aforementioned immigrant enterprises to recruit labourers by covering the costs up front, and the wages in the railroad building industry can easily cover the costs with a short-term contract that's compliant with the Anti-Coolie Act. This was called credit-ticket labourers and was the majority of Chinese labourers in America. The labourers would be indebted to the recruiters and pay off this debt in about a year or so, after which they became free men. The recruiters typically took a sizable cut: many would receive 150 dollars for a migrant, with a margin of 100%. In this period, emigration to the United States became very common across counties of the costal provinces, as the immigrant enterprises operated with incredible efficiency across the Pacific Ocean despite the lack of communication technologies, with many clans sending out men in each of their households to became formidable local powers. The trust in the system became so high that clan heads wagered entire fortunes on it. Of course, American Dream wasn't always achievable, and behind each story of "returning home with silk-laden clothes" there are tales of failure, mistreatment and even outright deaths which could bankrupt their families in China. But that would be another story to cover.

Here we finally get to the point to answer the question. The first way to send remittances: when migrants were recruited, they sometimes opt to add an advance of approx. 15-20 USD immediately paid to their clan. This was not very common though since they meant more fees to be paid later.

Once immigrants arrived to the US and paid off their debts, their wage became very, very desirable compared to their farming clan members in China. Some would opt to return to China for various reasons within a shorter time frame, and they would save up the money to pay off their own ship ticket, while any remaining amount would be quite a fortune to improve their own standings in the clan, marry a woman and build their families. It was the second way for remittances, an enviable endeavour but a dangerous one indeed. For those who were not too home sick and feared being robbed, many would stay for longer periods and send remittances via immigrant enterprises. Improvised banking institutions were established, who charged a fee to physically have people escort secured bundles of silver dollars and letters on scheduled ships, and deliver these bundles to the clans. This was called "银信", or "silver and letter", and was the majority of the remittances send from the US. Just like China's own banks, these immigrant institutions operated on basis of credit and trust: in a foreign and hostile land, being ousted for dishonesty would be the end of their careers.

And we have one final twist to the tale. Towards the end of 19th century, paper currencies were becoming the norm for the West. Now being paid silver certificates, virtually everyone had to exchange money via immigrant enterprises. This constituted the final empowerment to the migrant enterprises as they were increasingly involved in complex financial operations just like Qing China's domestic banks - which by the way were once dominated by Northerners and Yangtze River traders, so the entrance of once-overlooked Southern merchants represented a power shift. And with the opening of trans-pacific telegraph line, real-time communication became possible and it further aided in the operations of these companies, now becoming far-reaching enterprises. Remember the "remittance bureaus" I mentioned earlier, they began to be empowered via the transactions and became powerful institutions in local politics, whose impacts continue to influence China to this day. Many descendants of these institutions remain today as key business owners in both China and America.