I am a member of ResearchGate.net and I sometimes receive this 'full-text request' for some of my publications. Thus far, I generally provide them an access to my publication. However, I am not sure if this is good since the profile of the person who made the request is not always visible. Perhaps, some people are taking advantage of this somehow. Also, could this lead to any issues with the publishing organisation? Any ideas and recommendations?
First off: Receiving these requests means that people want to read your work. In nearly any academic scenario, this is a good thing. Embrace it, and help them! It's difficult to imagine what nefarious purpose could be served by somebody asking to read your paper for free, when they could read it anyway by doing an inter-library loan or paying a publisher $30. Some people consider ResearchGate itself to be nefarious; if this is you, you will have to choose between making your work more accessible and avoiding contribution to RG.
So, assuming that you want to help this person read your work, you need to check what you are allowed to do. Then,
If you are allowed to upload the final version of your paper, do that. This is probably only the case if you have paid the journal for so-called "Gold OA" and it has a permissive license.
If you are allowed to upload a post-review preprint ("postprint") to ResearchGate, do that. This question, and its answers, may be relevant here.
If you are not allowed to upload the paper to ResearchGate, it would be worth adding a note to that effect to the description of the publication, and (if you are allowed to upload it elsewhere, such as a personal website) giving a link to where it may be found. If contact details for the person who made the request are visible, you may wish to write to them privately and tell them where they can find it, or email them a copy.
Note that in addition to "this person wants to read your work" messages, ResearchGate occasionally sends "reminders" for any publications that don't have full-text uploaded asking you to provide it. You might want to follow step 1 or 2 above if applicable, but otherwise these are best ignored. Or, to stop the emails, you could upload a PDF that simply contains a statement of where the paper can be obtained from.
Work out if you can post pre-prints Many journals allow you to share some form of pre-print online. In some cases there is ambiguity as to whether posting pre-prints to ResearchGate is allowed by an agreement. For example, some publisher agreements limit sharing to personal websites, institutional repositories, etc. Nonetheless, most take-down requests that I've heard about on researchgate pertain to posting the publisher's formatted, full-text, version. Sherpa ROMEO provides general guidance about publisher policies.
Post your pre-prints in a range of forums. So, it makes sense that where possible you should provide the pre-print in a range of forums in order to facilitate the distribution of your work to those who do not have institutional access. There are many discipline specific repositories where you can post pre-prints.
Post the pre-print to ResearchGate. If you are on ResearchGate, then you have indicated a desire to use the system. So you should consider adding full-text pre-prints where you feel comfortable. That way, you wont get these requests because the full-text is already available.
I am also a member of ResearchGate and also sometimes get full text requests, as do my co-authors. What we do is if we are not able to email the scholar directly, we privately send a pre-print to the person asking (where we can) - usually with a message thanking them for their interest and an invitation to ask any questions should they arise.
You do have to be careful regarding the publisher's rules on this, look on their websites and if in doubt, ask.
You could request that the person make a formal request to your email account, if you are concerned that there is something strange about the individual not having their profile visible. Also, many journals (you may have to check with yours) will allow you to distribute an early version of the paper you published. For instance, a pdf of the last submission before acceptance. If you do not want to give the full, published paper out, then this is an option.
Most libraries have interlibrary loan, so I feel that it is somewhat lazy for a person to directly contact the author for a print, unless the request is from someone outside of academia or in a developing country. Those are the only reasons I have given out a full-text version of my paper, given that the request appears sincere.