A number of authoritative resources state that with few exceptions, water softeners and septic systems coexist without any trouble. For example, from the Water Quality Association:
Water softeners and septic systems are often found on the same property and, in the majority of these cases, no problems are indicated. Yet, there have been sporadic, mostly anecdotal reports of issues related to the combined use of these kinds of equipment at some sites. Experts in both fields are working together to better understand the interactions involved between water softeners and onsite wastewater systems. The recently completed study at Virginia Tech addresses this issue specifically.
In other words, could you have a problem? Yes, it's possible. Others have. But the likelihood is extremely low.
It's not clear what articles you've found on the web that argue the opposite. But the above reference includes a rigorous study performed by researchers at Virginia Tech, so I find it highly credible. Other sources on the Internet may not be as trustworthy.
For what it's worth, there isn't salt per se in the filtered water. While NaCl is the input for the water softening system (for those that use salt…there have been attempts to create water softening systems that don't use salt), by the time the water gets to the domestic tap, that salt has been split into ions, which react to remove the hard water ions (mainly calcium and magnesium) from the water.
The water does have sodium by itself, and that can be a health issue for some people. But the overall concentration in the domestic water supply is not consequential for the septic system.
The bigger issue would be the concentrated salt solution that comes from backwashing the softening media. This solution is not great wherever it's discharged, but note that there's no requirement that it be dumped into the septic tanks. So if you're really worried, you could find somewhere else to direct that flow.
That said, from the same reference as above:
…well operated softeners at their highly rated efficiencies of 3000 to 4000 grains of hardness per pound of salt used will not harm – and could even aid – septic systems
In other words, problems generally appear when the water softener isn't properly configured, and is using more salt than necessary for water treatment. When correctly configured, its use can in fact help the septic system work better.