When multiple aura:if are nested together, I'm getting an error like the following:

Access Check Failed! AttributeSet.get(): attribute 'value2' of component 'markup://c:demo1 1:0'
is not visible to 'markup://aura:if 6:0'.

Example Reproduction

<aura:application >
 <aura:attribute name="value1" type="String" access="private" />
 <aura:attribute name="value2" type="String" access="private" />
 <lightning:input value="!v.value1" label="Value 1" placeholder="Enter 'a'" />
 <aura:if isTrue="!v.value1 eq 'a'">
 <lightning:input value="!v.value2" label="Value 2" placeholder="Enter 'b'" />
 <aura:if isTrue="!v.value2 eq 'b'">
 <lightning:input label="Value 3" value="!join(' ',v.value1,v.value2)" />
 </aura:if>
 </aura:if>
</aura:application>

This occurs when the attribute is private, and there's at least two levels of aura:if statements. It also only occurs after causing the innermost lightning:input to render, then subsequently unrendering it by changing Value 1 a few times. Once this happens, all further renderings will cause this error.

Why is this happening, and what can I do to avoid it?

This is a bug with access checking in the Lightning Web Component framework, and may manifest as an error in Aura components when this happens. There are a few different ways to reproduce it; the one in the answer is just one form. The known solutions are as follows:

Make The Attribute Public

The access exception will only occur on private attributes.

Avoid aura:if

The way aura:if rerenders causes the bug to surface in LWC. Instead, hide the elements using normal CSS (or SLDS). Here's an example fix:

<aura:application extends="force:slds">
 <aura:attribute name="value1" type="String" access="private" />
 <aura:attribute name="value2" type="String" access="private" />
 <lightning:input value="!v.value1" label="Value 1" placeholder="Enter 'a'" />
 <div class="!v.value1 eq 'a'?'':'slds-hide'">
 <lightning:input value="!v.value2" label="Value 2" placeholder="Enter 'b'" />
 <div class="!v.value2 eq 'b'?'':'slds-hide'">
 <lightning:input label="Value 3" value="!join(' ',v.value1,v.value2)" />
 </div>
 </div>
</aura:application>

Avoid Usage of LWC Components

Using any component that is a LWC component (basically anything in the lightning namespace) can result in this error. You can replace it with the basic ui namespace component (e.g. ui:inputText instead of lightning:inputText). This will probably not be practical in most applications, but it is a possible workaround.

This has been filed as W-5722782.