How can I create non-forced (natural) engagement from my customers / fans to share and refer to other friends?

I know this is a tricky problem to solve on optimizing costs and trying to get customers refer your service.
Referral schemes work perfectly where incremental costs of adding additional service is marginal, like the case of DropBox.
When the incremental costs are a significant portion and adding additional service is going to impact your CapEx, you need to identify the portions of the service which can be given away with less cost. So in this example, how about a free last-mile delivery for the next shipment for every friend who refereed your service to their friends? This won't hurt your bottom line a lot but brings up the top line.
Another way is leveraging the social angle associated. When someone is shipping a good internationally, the next thing he want to do is tell people about their sharing. Can you actually integrate social channels to your service, so that they can share it with world easily.

Hi, great question! got my brain working over here!
FYI, I'm a growth hacker so I'll try to actually be of use to you here.

First thing that comes to mind is that you might be going about this wrong. Is common for most of us to have a biased researched online to solve a problem we are having, our search is biased because of something we might have heard of in the past so we start with that. When we find the answers we were looking for regarding that biased search we stick to that and get stuck trying to figure out how it applies to us and make it work. In business this is very very common, that’s why a lot of us are here, and a lot of startups need to pivot once they realize that they got pulled into thinking their prior focus what the right one.

You start off asking how to engage more consumers, then quickly moved to a specific strategy that worked for other companies. The truth is that for these companies these tactics were only 1 thing from a variety of efforts. Which is why I think you might be focusing on the wrong one.

Giving away freebies works, but that’s just traditional marketing coupled with their growth hacking techniques. It doesn't work for you, fine.
The reason why your consumers are not sharing as much as they are, or raving to their friends is because they are simply users – what you need is hardcore aficionados in something that you provide.
How? Improve your business development and marketing by super targeting your demographic. Find a subset of your clients who are hard core about what they buy, possibly hard core about international purchases, or a large group who buys the most popular item through your site.
How? Build a persona. You have probably heard this, but is extremely important. {I have continued my response to you through my blog, due to length... please continue reading here: http://UnthinkEverything.blogspot.com }

I have some examples and stuff I hope you enjoy and find useful.

Digital marketers that serve customer-centric organizations understand what customers want and rely on customer data to capture customer insights and share this across the organization. Brands must be committed to customer-centricity concentrate on what their clientele wants and needs, and develop new products and services based on that. To win in today’s media mix, the customer dictates reach, frequency, and how they buy. Just how your customers and prospects are being engaged is critical to sustained success in the long term. More and more companies, who have relied on email marketing as the default method for communications, are now wondering if a better, more engaging way exists. Businesses must be keen to align with evolving customer requirements and need a new content writing strategy for more natural “ask an assistant” searches. Your business will benefit if you can view things from the customer's perspective, and not from an internal business standpoint. With the demise of traditional inbound and outbound marketing efforts, the need to find a bulls-eye marketing plan starts with a better understanding of your customer. After the right content is created, schema mark-up helps to match content to user queries, and then the many forms of marketing that piece is needed. Predictive guesses vary widely but the common consensus is that it is five times more expensive to gain a new customer as to retain an existing on fore.
If you are a customer-centric company that wants to engage your customers in more mutually beneficial ways, it will require co-operation from design teams, sales, and customer support for brand alignment and ultimate success. To accomplish what is needed, entire teams should look at where your business is currently at in terms of rankings, visual search trends, and user satisfaction. It may seem like the tables have turned but watch your metrics and by focussing on the user's intent and needs first, those should follow. In the future, customer engagement strategies must go deeper than just to creating content to maintain strength and positioning. Real data that leads design decisions will better satisfy the user experience than fancy widgets and flashy design elements.
A consumer-centric approach includes identifying the questions prospective customers are asking, how they seek answers, and how to use those insights to shape your content strategy. Along with the changes in the search algorithms, devices, and competitive landscapes, search marketers must respond and revise according to the fluid ways users chose to use search. That may be voice, text, desktop, apps, query length changes, channels, question formats, and user expectations. Maybe that means switching to video content on YouTube, or developing your Google Assistant page, or being more visible on Google Maps, as well as social platforms.
Preparing your communications to align with the voice of the customer helps you improve your services. Customer feedback is an especially important tool. It may alert a business of future communication trends.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath