You may employ a gentile to do a job on moveables and be paid a fixed amount upon completion. If he works on Shabbos to get ahead, it is his own concern because you don’t mind if it will be completed a day later. The fact that the work is being done for a Jew should be kept discreet in order to avoid maris ho’ayin. If some Jews know, it's okay. (ShA1 – MB1,2,4)
Chazal were more machmir when the job is attached to the ground. Everyone knows it's Jewish-owned property. Although he is being paid for the job, it entails maris ho’ayin and is therefore forbidden. People may think that the gentile is being paid for the day. For example, he is building you a wall or harvesting your field within the techoom of the city. This is forbidden even if you are the only Jew in the city. (ShA1 – Rama1 – MB4)
Even if the gentile is merely preparing building materials on his own premises before the actual building, it is ossur on Shabbos. It is still called “melleches Yisroel” and it’s for something that will attach to the ground. (ShA2 – MB14)
The above prohibition applies only when the materials belong to the Jew. Some are lenient if people don’t know that they belong to the Jew. But if the wages include that the builder provides the materials, he can work on his own premises on Shabbos even if everyone knows that the work is for a Jew. (Rama2 – MB14)
It is forbidden for a goy to do public melachos that people know are being done for a Jew on Shabbos, even if it isn’t attached to the ground (e.g. ship making). There is maris ho’ayin just like building. (ShA4)
In cases of maris ho’ayin you must protest against the gentile worker and get him to stop even if you originally stipulated with him not to work on Shabbos. (MB15, 22)
If you hired a gentile scribe or weaver for a whole year to work on jobs that you will need – can he work for you on Shabbos? Not if you calculate the days he takes off. And not if he is working on your premises. Otherwise, there is still an argument if the scribe can work for you on Shabbos. The Elya Rabba rules leniently, but it is better to fulfill the strict opinion. (ShA5-BeH”v’yaish”)
Let’s say you’ve hired a gentile maid for an extended period of time to perform any variety of tasks. She may not do any melacha for you on Shabbos even without your request and even on her own premises. To perform melacha for herself on your premises is allowed when it is evident that it is for herself (e.g. sewing her blouse). (MB30)
One method is to stipulate at the outset of the partnership that you aren’t expected to work at all on Shabbos. This will avoid the partner being deemed as your shaliach. You tell him: “Work on Shabbos and take the entire day’s profit for yourself no matter how much. I will work by myself on a weekday and keep all the profits of that day for myself.” It’s like you are not partners for those two days. (ShA1 – MB intro., 2,4 – BeH “sheyaish”)
When you come to split the profits, the gentile partner can choose just to split it all equally even if the profits from Shabbos exceed the profits that you made on your day. Some poskim forbid you to suggest this yourself. (ShA2 – MB13)
If it will not be known, or you lost track of how much profit each day brought in, you can demand to split it equally. (MB6 – ShaH5)
Here are two cases which are forbidden:
The stipulation at the start was just: “You work on Shabbos and I’ll make up for you one day during the week.” Your extra work is like payment for him being your shaliach to work on Shabbos.
You said nothing until dividing the profits, “You take money for your work on Shabbos and I will take for just the weekday.” (MB intro.,7)
If no stipulation was made, you can allow the business to operate on Shabbos. Just at the division of profits, subtract the amount made on Shabbos first. If the amount is unknown, subtract a seventh for the partner and split the remainder equally. If by subtracting the profits of Shabbos (or 1/7) you will suffer a great financial loss, you can rely on the opinion that holds it’s okay to split it all equally. (ShA1)
If the nature of your business is for people to come and help themselves doing the melacha so that your gentile partner does no work, you are allowed to split the Shabbos earnings equally. One consideration: don’t divide only those earnings at one time – that will be s’char Shabbos. (MB1)
Even if you didn’t make the stipulation with your partner at the start of your partnership, it’s not too late. Look at the Shulchan Aruch for a way to restart the business with the stipulation. (ShA3)
Another option is to rent out part of your share to another gentile (or to your existing partner). For example: He will pay you for 55 days’ worth of using the business and he will work the 52 Shabboses, plus a few extra weekdays so it won’t be S’char Shabbos. (Rama3 – MB15)
You are allowed to give your money to a gentile to do business with it also on Shabbos. You will get a percentage of the profits. (ShA4)
You are allowed to hire a gentile at a fixed wage to sell merchandise for you even on Shabbos, as long as you don’t tell him “sell on Shabbos”. If market day is only on Shabbos, then it is ossur because it’s as if you tell him “sell on Shabbos”. (ShA5 – MB21)
If you have an appliance (e.g. oven) on your property and non-Jews come to use it on Shabbos (even without your permission) any payment that they want to give you will be ossur because of s’char Shabbos. (ShA6 – MB24)